INCLUDITO  SQUAEE  BOOT,  CUBE  BOOT,  AND  OTHER  BOOTS. 


A  HIGHLY  PRACTICAL,  BRIEF  AND  UNIQUE  METHOD 

FOR    THE    EXTRACTION    OF    ALL 

ARITHMETICAL  ROOTS. 


A  SCIENTIFIC  PROCESS 

NOT  HERETOFORE  PRESENTED  IN  ANY  PUBLISHED  WORK 

ON  ARITHMETIC,  AND 

SAVING  NINE-TENTHS  OF  THE  LABOR 

USUALLY  NECESSARY  FOR  THE 

EXTRACTION  OF  ROOTS,  AND  ESPECIALLY  OF  CUBE  ROOT, 

UNDER  THE  RULES  NOW  EMPLOYED. 


FOR   THE   USE    OF    ALL    GRADES    AND    ALL    SCHOOLS 

ABOVE  THE  PRIMARY,  AND  FOR  TEACHERS 

IN  PARTICULAR. 


^^  OF  THB***^ 


"  OF  THE 

lUKIVEESITY] 


— BY— 

G.  D.  HINES,  A.  M. 
w 


CLEVELAND.  0.: 
J.  R.  HOLCOMB  &  CO.,  PUBLISHERS, 

1886. 


COPYBIQHT,   BY  J.   E,   HOLCOMB  &  Co.,   1886. 

,1^  -Copies  of  this  work  will  be  sent  to  any  address,  postpaid,  on  receipt  of  price.    Active 
Agents  wanted.  J.  R-  HOLCOMB  &  CO.,  Pdblishbes, 

Cleveland,  Ohio. 


^7-9^ 


DEDICATION. 


TO  MY  WIFE, 

THE  SYMPATHETIC  SHARER  WITH    ME    OF  THE  MIXED 

CUP  OF   FORTUNE   INCIDENT  TO 

A   LONG   SCHOOL  LIFE,   THIS   LITTLE  VOLUME 

IS  AFFECTIONATELY   INSCRIBED. 

G.   D.   HiNES. 


PREFACE. 

-^HIS  little  work  needs  but  a  short  introduction.  It  is  brevity  itself,  and  whatever  of  merit  it  may 
possess,  is  largely  due  to  its  brevity  and  practicability.  The  methods  of  treating  difficult  roots, 
herein  contained,  and  especially  of  cube  root,  were  first  suggested  to  the  Author  in  the  winter 
of '8i  and  '82,  while  teaching  the  Lincoln  School,  Plumas  County,  California.  The  pupils  of 
that  school,  as  is  usual  with  most  pupils,  on  first  meeting  the  subject,  were  having  trouble  with 
cube  root.  This  caused  the  teacher  to  put  his  wits  to  work  in  the  almost  hopeless  effort  to 
devise,  if  possible,  some  easier  and  shorter  way  of  solution  than  the  prolix  processes  extant  in  the 
arithmetics ;  and,  after  some  days  of  close  scrutiny  of  the  meaning  and  relation  of  roots  and  powers, 
the  Author  detected,  for  the  first  time,  a  new  method  of  teaching  cube  root.  The  treatment  of  other 
roots,  and  of  surds,  naturally  followed ;  and  the  result  has  been  what  my  fellow-teachers  and  others 
will  find  in  the  following  pages  of  this  work. 

It  is  believed  that  the  addenda  of  Interest,  with  its  unique,  brief,  and  simple  treatment,  will  prove 
an  attractive  feature  of  the  book. 

This  work  is  distinctly  original.  It  details  our  own  discoveries  and  is  the  product  of  our  own 
thought  respecting  the  treatment  of  that  difficult  subject,  cube  root.  We  send  it  forth  on  its  mission, 
conscious  that  it  must  stand  or  fall  on  its  own  merits.  We  are  fully  persuaded,  at  the  same  time,  that 
it  needs  only  to  be  seen  and  understood  to  be  appreciated ;  and  that,  if  generally  introduced,  it  must 
supplant  the  perplexing  and  unsatisfactory  rules  in  the  text-books  on  cube  root  and  the  higher  roots, 
and  on  surds.  We  ask  an  impartial  examination  by  our  fellow-teachers  everywhere,  and  believe  they 
will  find  the  little  book  helpful,  if  not  indispensable,  to  them. 
Selma,  Cal.,  May  28,  1886. 


cube"root 

AND  OTHER  ROOTS  UPROOTED. 


Def. — A  root  is  one  of  the  equal  factors  that  has  been  repeated  in  multiplication  a  given  number 
of  times,  to  produce  a  given  pov/er.  Ex. — 3  is  the  cube  root  of  27,  having  been  twice  repeated  in 
multiplication,  or  thrice  used  as  a  factor,  to  produce  the  power.  27  is  the  cube  root  of  19683,  having 
been  twice  repeated  in  multiplication,  or  thrice  used  as  a  factor,  to  produce  the  power.  15  is  the  fourth 
root  of  50625,  having  been  three  times  repeated  in  multiplication,  or  four  times  used  as  a  factor,  to 
produce  the  power.  And  so  on  for  any  other  roots,  integral  or  fractional,  positive  or  negative.  To 
find  the  roots  of  all  powers  that  legitimately  belong  to  arithmetic,  is  the  chief  object  of  this  work. 

Obs. — As  this  treatise  deals  chiefly  with  cube  root  and  other  higher  roots,  no  extended  notice  of 
square  root  will  be  taken.     Only  an  incidental  usage  will  be  made  of  it. 

THE  BASIS. 

It  is  a  well-known  fact  that  the  principles  underlying  Arithmetical  evolution  are  derived  from  the 
mother  science  of  Algebra,  and  that  the  arithmetical  rules  have  been  formulated  out  of  the  algebraic 
formulce.  No  other  rules  for  the  extraction  of  roots  have  been  presented  in  the  arithmetics,  and  per- 
haps, substantially,  no  others  can  be  framed  than  those  which  depend  on  algebraic  principles.  But 
certain  rules  can,  nevertheless,  be  formulated,  which,  while  they  have  reference  to  algebraic  principles, 
completely  revolutionize  the  old  mammoth  rules,  and,  in  brevity,  almost  annul  them. 

THE  NEW  TREATMENT. 

The  methods  about  to  be  illustrated  neted  have  no  reference  to  algebra,  nor  do  they  require  any 
knowledge  of  that  science.  They  annihilate  the  "cubic  block"  system,  which  clearly  presents  the 
principles  of  evolution  to  only  the  maturer  scholars ;  and  then  only  in  cube  root,  and  are  of  no  real 
advantage  in  the  actual  work  of  even  the  cube  root,  in  very  large  numbers,  and  certainly  of  no  advant- 
age in  extracting  any  other  than  the  cube  root. 

CUBE  ROOT. 

We  will  now  present  our  method  for  the  extraction  of  the  cube  root.  Powers  are  either  perfect  or 
imperfect.    15625  is  a  perfect  cube,  while  18740  is  an  imperfect  cube,  or  third  power,  and  is  called  a  surd. 

We  will  present  a  rule  for  the  extraction  of  the  cube  root  of  perfect  third  powers,  and  one  also  for 
that  of  surds. 

Obs. — It  may  be  remarked  that  a  surd  may  be  considered  an  imperfect  power  of  any  degree  what- 
ever. Thus,  18740  may  be  considered  an  imjaerfect  square,  cube,  fourth  power,  or  any  other  power ; 
for  we  may  require  the  approximate  square  root,  cube  root,  fourth  root,  or  any  other  root,  of  18740. 
But  it  sometimes  happens  that  a  perfect  power  of  one  degree  is  a  surd  of  another  degree,  and  vice  versa. 

Ex. — 25  is  a  perfect  square,  but  an  imperfect  cube ;  while  27  is  an  imperfect  square,  but  a  perfect 
cube, 

EXACT  CUBES  OF  TWO  PERIODS. 

Let  us  extract  the  cube  root  of  the  following  numbers,  viz.: 


-^  13824 
74088 


185 193  A  little  observation  and  practice  enable  us  to   determine  by  inspection  the  root 

250047  figure  of  the  first  period  of  the  pow^r.  Thus,  the  root  figure  of  13,  in  the  first  of 
262144  91 125  the  preceding  numbers,  is  2.  And  by  the  new  method,  the  root  figure  of  the  last 
166375  ■  97336  period  is  4,  when  the  period  ends  in  4.  Hence,  the  cube  root  of  13824  is  24.  The 
704969  226981  root  figure  of  the  first  period  of  74088  is  4,  and  the  root  figure  of  the  last  perlod'is  2, 
when  the  period  ends  in  8,  (It  is  8  when  the  periocl  ends  in  2.)  So  the  cube  i-oot,  of  the  last  number 
is  42.  The  cube  root  of  262144  is  obtained  in  the  same  way.  The  root  figure  of  262  is  6,  and  of  144  it 
is  4.  So  the  ^262144  is  64.  Again,  the  root  figure  of  the  first  period  of  166375  is  5;  and  the  root 
figure  of  the  last  period  is  5,  when  the  period  ends  in  5.  So  the  1^166375  is  55.  704969  gives,  for  the 
first  root  figure,  8 ;  and  the  last  is  9,  when  the  period  ends  in  9.  So,  the  ^'704969  is  89.  In  like  man- 
ner, 185193  gives,  for  the  first  root  figure,  5  ;  and  last  figure  is  7,  when  the  period  ends  in  3.  (It  is  3, 
when  the  period  ends  in  7.)     Thus,  the   1^185193  is  57.     The  cube  root  of  250047,  for  reasons  already 


MATHEMATICAL   ROOTS   UPROOTED. 


Stated,  is  63.  The  cube  root  of  9"25,  for  similar  reasons,  is  45.  The  ^97336  is  to  be  written  out 
impromptu,  just  as  the  previous  roots  have  been  ;  making  the  last  root  figure  6,  when  the  final  period 
ends  in  ^.  Also,  the  ^226981  is  61,  the  last  root  figure  being  i,  when  the  concluding  period  ends  in 
I.  Observe  that,  in  all  perfect  cubes,  the  final  root  figure  is  simply  chosen,  according  to  the  character 
of  the  terminating  figure  of  the  power.  This  is  a  great  saving  of  time  and  work,  as  will  be  shown 
hereafter. 

EXPLANATION. 

The  reasons  for  the  foregoing  selection  of  the  final  root  figure,  depend  on  a  very  plain  principle.  In 
all  perfect  third  powers,  it  is  evident  that  the  final  figure  of  the  pnver  arises  from  the  multiplication 
of  the  final  figure  of  the  root  tzuice  into  itself.  Now,  if  we  multiply  the  nine  digits,  respectively,  twice 
into  themselves,  we  shall  have  this  result,  viz.: 

Observing  the  final  figures  of  these  powers,  we  see  that  the  final  root  figure  i  pro- 
duces a  1,  on  being  twice  multiplied  into  itself.  We  see  that  the  final  root  figure  2 
produces  an  8,  and  an  8  a  2  ;  that  3  produces  a  7,  and  a  7  a  3  ;  that  a  4  produces  a 
4;  that  a  5  produces  a  5 ;  that  a  6  produces  a  6,  and  that  a  9  produces  a  9. 

Obs. — The  cubes  of  the  several  digits,  viz.:  i,  8,  27,  64,  125,  216,  343,  512,  729, 
must  be  so  thoroughly  familiar  to  the  student  that  he  can  select  the  root  figure  of  the 
first  period  impromptu. 

If  the  first  period  is,  in  magnitude,  between  i  and  8,  the  root  figure  is  i ;  if  the 
first  period  is,  in  magnitude,  between  8  and  27,  the  root  figure  is  2  ;  if  the  first  period 
is  embraced  between  27  and  64,  the  root  figure  is  3 ;  if  it  is  between  64  and  125,  the  root  figure  is  4; 
and  so  on. 

1.  What  is  the  root  figure  of  the  first  period,  if  the  period  is  comprised  between  512  and  729? 

2.  What  is  the  root  figure  of  the  first  period,  if  the  period  is  larger  than  729  ? 

EXAMPLES. 


iXiX 1=  I 
2x2x2=  8 
3X3X3=  27 
4X4X4=  64 
5x5x5  =  125 
6X6X6  =  216 

7  X  7  X  7  =  343 
8x8x8=512 
9  X  9  X  9  =  729 


Let  it  be  required  to  extract  the  cube  root  of  the  following,  by  the  foregoing  principles,  viz.: 

Thus,  we  may  write  out,  impromptu,  according  to  the  foregoing  principles,  the'roots 
of  these,  and  of  all  perfect  cubes  involving  only  two  periods. 

Let  the  student  find  the  roots  of  the  following,  writing  out  the  answers,  off-hand,  with- 
out any  figuring  or  formal  extraction,  and  choosing,  at  sight,  the  final  root  figure,  ac- 
cording to  the  character  of  the  terminal  figure  of  the  power,  viz.: 


39304 
#"■091125 
f  46656 
f  405224 
f  389017 
f  474582 


lA 


]^.  0000021744 
^.000024389 
f^.  000079507 
^.000614125 
f  2.197000 
f^  941. 192000 


0l>s. — The  last  six  preceding  numbers  consist  of  three  periods,  but  may  be  solved  without  premed- 
itation by  the  same  rules,  respecting  the  terminal  figure,  as  are  given  for  powers  of  two  periods. 

EXACT  CUBES  OF  THREE  PERIODS. 

Let  us  extract  the  cube  root  of  the  following  numbers: 

:     .-  7.    153990656 

V'^^.  I.       44361864  4-     12.812904  8.      387420489 

\  Q^^  2.     134217728  5.        5545233  9-       10077.696 

\    ,        ',f       3.       12812904  6.        2000376  10,       36.926037 

^/u  '     •'   Taking  the  first  of  the  preceding  numbers,  and  selecting  the  first  root  figure,  3,  we  take  its  cube 
44361864  I  354.   I  from  the  period,  and  to  the  remainder  attach  the  next  period.     Find  the  second 
/^  .       27  I  Ans.     root  figure  by  dividing  the  partial  dividend,  save  the  two  right-hand  figures,  by 

triple  the  square  of  the  first  root  figure.     Choose  the  last  figure  according  to  the 

17.361  I  character  of  the  terminal  figure  of  the  power. 


27 


75 


In  the  same  manner,  the  cube  root  of 
134)217728  is  1  512, 
125  I  Ans 


92,17 
So,  the.  cube  root  of 


For  a  partial  divisor  we  take  three  times  the  square  of  the  first  root  figure, 
and  again  choose  the  last  figure. 


1 28 1 2004  is  I  234,  found 


thus:     Partial  divisor  12      48,12  The  last  figure  is  simply  chosen. 

The  cube  root  of  12:812904  is  2,34,  the  same  in  form  but  different  in  value. 


MATHEMATICAL   ROOTS   UPROOTED. 


{  I 


3  I  45-45        Dividing  and  allowing  for  a  completed  divisor,  we  get  7  for  the  second  root  figure, 
and  choose  the  last,  which  must  be  7.     Why? 

The  ^2000376=  126.     Ans.  /  -^>  •-  f    ' 

Partial  divisor      3 


!4- 


10.00  Why  is  the  last  root  figure  6? 

No  other  figuring  than  the  above  is  necessary. 


Divisor  52x3=75 


The  ^153990656=536. 
125 

Always  triple  the  square  of  first  root  figure  for  ^pktt&l  divisor  of  all 


289.90 


but  the  two  right-hand  figures  of  the  partial  dividend. 
The  ^387420489  =  729. 
I  343 

Divisor      147  |    444.20  Having  obtained  the  second  root  figure,  we  choose  the  last  unerringly. 

Why  is  it  9  ?     How  is  147  obtained  ? 
21.6. 


The  ^10077.696: 
I    8 

Divisor        12  |    20.77        O"^  the  same  principles   the  cube  root  of 


36.926037 
27 

99.26 


3-33. 


and  is  found  thus :  27 

On  the  law  of  the  terminal  figure  of  the  power  depends  the  secret  of  this  new  method  with  cube 
root.  It  is  worth  much  to  the  student.  Let  him  verify  all  of  the  above  answers,  by  going  through  the 
actual  work  in  every  case,  and  thus  acquire  the  needed  familiarity. 


THIRD  POWERS. 


K 

^^K         Involving  periods  of  noughts  at  the  left  or  right  of  the  significant  figures,  and  extracted  by  the 
^^B.  foregoing  rule. 

i 


The  ^.000520476129  =  .0809. 
512 


Partial  divisor     192  174-75  The  partial  divisor  is  not  contained  in  the  brought-down 

dividend  shorn  of  its  two  right-hand  figures,  and  we  place  a  nought  for  the  third  figure  of  the  root,  and 
arbitrarily  choose  the  last  figure  which  is  9. 

The  ^.048228544000  =  .3640. 
27 


Divisor 


27 


212.28 


Dividing,  we  find  the  partial  divisor  is  contained  in  212  only  6 


^ times,  allowing  for  the  effect  of  a  finished  divisor.  '  We  choose  the  remainder  of  the  root  figures. 

.0192. 


The  ^.000,007,077, 
I 


Divisor  3  60.77  Dividing  by  the  partial  divisor,  we  find  it  will  go  into  the  par- 

tial dividend  the  largest  possible  number  of  times.     (No  divisor  can  go  more  than  9  times.)     The  last 
figure  of  the  root  is  2.     Why  ? 


The  ^.000618470208: 
512 


Divisor 


192 


.0852.     Ans. 


1064.70 

I.     How  is  the  partial  divisor,  192,  obtained? 
We  select,  unerringly,  the  last  root  figure. 


MATHEMATICAL   ROOTS   UPROOTED. 


ADDITIONAL  EXAMPLES  FOR  THE  STUDENT. 

Find  the  cube  root  of  the  following,  viz.: 

Find  the  value,  also,  of  these  : 

I.       84604519. 

7.     f       13481272 

2.           2803221.                                ^ 

8.     1^8615.125000 

3-        3176523. 

9-     f  738.763264 

4.     382657176. 

10.     f   561.515625 

5.     40.353607. 

II.     -^      21024576 

6.         1520875. 

12.     if      67917312 

In  solving  these  examples,  let  it  be  understood  that  our  only  rule  for  cubes  of  three  periods  is,  to 
take  out  by  inspection  the  root  figure  of  the  first  period,  and,  having  taken  its  cube  from  that  period 
and  attached  the  next  peiiod  to  the  remainder  for  a  new  dividend,  to  find  the  next  root  figure  by  divid- 
ing the  partial  dividend  by  triple  the  square  of  the  first  root  figure,  and  arbitrarily  ckoose  the  last  figure 
of  the  root,  according  to  principles  already  explained. 


Perfect  Cubes  of  More  Than  Three  Periods. 


We  will  now  extract  the  cube  root  of  some  numbers  of  more  than  three  periods,  and  show  that  the 
new  method  applies  to  them,  with  a  very  slight  amount  of  additional  work. 

Let  it  be  required  to  extract  the  cube  root  of  the  following  numbers,  viz.:  8024024008,  10460353203, 
98867482624,  1 226 1 5327232,  1 54480441 6.  Taking  the  first  of  the  above  numbers,  we  proceed  as  with 
powers  of  three  periods,  thus  : 

8,024,024,008  I  2002.     Ans, 


Divisor  12.00  24.024  Explajtation.—Ymdimg  the  first  root  figure,  deducting  its  cube,  and  at- 

taching the  next  period  for  a  dividend,  we  find  that  the  partial  divisor,  12,  is  not  contained  in  the  par- 
tial dividend,  24,  of  the  second  period,  and  we  attach  the  next  period.  The  root  thus  far  found  is  20, 
and  triple  its  square  is  1200,  the  next  partial  divisor,  which  is  again  not  contained  in  the  brought-down 
dividend  shorn  of  the  two  right-hand  figures,  and  the  root  now  found  is  200,  and  we  choose  the  last 
figure. 

The     ■ 


12.6. 1 


10460353203 
8 

24.60 
1261 


2187.     Ans. 


212X3=1323         11993.53  Explanation. — We  divide,  as  usual,  the  first  partial  dividend  by 

the  triple  squai-e  of  the  first  root  figure,  which  is  12,  and  obtain  i  for  the  second  root  figure.  We  finish 
the  divisor,  12,  by  adding  to  it,  successively,  advanced  one  place  to  the  right  for  each  addition,  the 
triple  product  of  the  two  root  figures,  and  the  square  of  the  last  One.  This  finished  divisor  we  multiply 
by  the  last  root  figure,  and  take  the  product  from  the  brought-down  dividend.  We  need  only  a  second 
partial  divisor,  the  triple  square  of  21,  to  find  the  third  root  figure,  and  we  choose  the  last  figure  of 
the  root,  according  to  the  character  of  the  terminal  figure  of  the  power. 

4624. 


The  ^98867482624  = 
64 


Ans. 


Divisor 


48.36 
72 


348.67 
33336 


Fin.  div.         5556        15314.82  Using  the  same  finished  divisor  as  an  approximate  divisor,  we  find 

the  next  root  figure  to  be  2,  and  then  we  select  the  last.    Observe  that  36  is  put  in  the  niche  of  the  other 
two  parts  of  the  divisor,  to  save  space. 

Obs. — To  insure  accuracy  in  finding  the  third  root  figure,  it  is  generally  best  to  take  for  a  divisor  the 
triple  square  of  the  first  two  root  figures.  Even  then  there  is  an  immense  saving  of  work,  time  and 
space  over  the  old  methods.  By  the  modes  of  solution  practiced  heretofore,  as  treated  in  the  books,  the 
work  of  the  above  example  is  absolutely  overwhelming,  covering,  with  the  rule  and  the  explanation,  from 
three  to  five  pages.  Let  the  pupil,  for  the  present,  accept  on  trust  such  parts  or  features  of  the  rule  as 
he  may  not  thoroughly  understand.     Further  elucidation  will  be  given  in  due  time. 


MATHEMATICAL   ROOTS   UPROOTED. 


1st  Divisor 


48.81 


^122615327232  =  4968. 
64 

586.15 
53649 


5961 


49663.27 
43218 


2nd  Divisor       49''X  3  =  7203  .  ,    ,.  . 

6445  It  is  only  necessary  to  frame  a  second  partial  divisor 

to  obtain  the  third  root  figure,  and  then  we  arbitrarily  choose  the  last  figure  of  the  root.     What  have 
we  saved  by  this  abbreviated  method  ? 

We  have  saved  the  completion  of  the  second  divisor,  the  formation  of  the  third,  the  completion  of 
the  third  divisor,  and  all  the  multiplications  and  subtractions  connected  with  these  last  periods,  the 
prolixity  and  difficulty  of  which  rapidly  increase,  under  old  methods,  as  we  approach  the  end. 

The  1^1544804416  =  1156.     Ans. 
1st  divisor  3         I 


Finished  divisor  331 
2d  divisor  363 


544 
331 


2138.04 

.1815  There  is  no  necessity  for  multiplying  the  second  partial  divisor,  363, 

by  the  third  root  figure,  5.     We  do  so  here,  to  show  that  the  remainder  of  the  dividend  divided,  is  less 
than  the  divisor. 

The  ^12,521,107,822,861  =23221.     Ans. 
1st  divisor       12 


Fin.  divisor 
2nd  divisor 


1389 


45.21 
4167 


Explanation. — We  complete  the  first  partial  divisor,  and  take 
its  product,  with  the  corresponding  root  figure,  from  the  l^rought- 

I3174  down  dividend.  The  triple  of  the  square  of  the  root  now  found 
is  a  partial  divisor  by  which  all  the  other  root  figures  may  be 
367  found,  except  the  last,  and  that  is  simply  chosen.  Having  framed 
317  the  partial  divisor,  1587,  we  ascertain  how  often  it  is  contained  in 
the  corresponding  partial  dividend  (always  excepting  the  two 
50  right-hand  figures),  and  multiply  the  divisor  by  the  quotient  fig- 
ire,  and  subtract  the  result  from  the  portion  of  the  dividend  divided,  as  in  ordinary  division.  Having 
ound  the  third  figure  of  the  root,  we  use  for  a  dividend  the  remainder  of  the  last  partial  dividend,  and 
lor  a  divisor  we  use  the  previous  divisor  shorn  of  its  right-hand  figure.  And,  in  multiplying  this  divisor 
by  the  quotient  figure,  we  reckon  in  the  number  of  units  that  would  be  to  carry  from  the  figure  cut  off. 
If  there  were  more  periods  than  five,  the  process  might  be  continued,  by  dropping  figures,  successively, 
from  the  right  of  the  divisor.  But  the  last  figure  of  the  root  is  always  chosen.  And,  now,  in  what  is 
said  above,  respecting  the  finding  of  some  of  the  root  figures  by  ordinary  divison,  lies  the  germ  of  the 
method  herein  treated,  for  the  approximate  extraction  of  the  cube  root  of  surds,  to  be  explained  in 
due  time. 

Take  one  more  example  involving  five  periods.     Let  it  be  required  to  extract  the  cube  root  of  the 
number 

599)183,710,672,625  1  84305.     Ans. 
512 


1st  divisor  192.16 

20176 

Bsor  842x3  =  2116.8 

I 

oot  fieure.  which,  fror 


871.83 
80704 


Elucidation.— E-aixTiCt  in    the  usual  way  to  find  the 

first  two  figures  of  the  root.     Then,  as  will  be  seen  by  in- 

64797.10  specting   the  work,   there  is  a  necessity  for  finishing  only 

63504  one  partial  divisor,   and    afterward  forming    another  one 

from  the  first  two  root  figures.    This  second  partial  divisor 

1293  we  use  to  find  two  more  root  figures,  and  then  choose  the 

figure,  which,  from  the  character  of  the  terminal  figure  of  the  power,  must  be  5. 
Ods.—ln  the  above  solution,  there  is  an  immense  saving  of  labor,  time,  and  space.  The  student 
cannot  reahze  the  wide  difference,  if  he  has  not  gone  through  the  labyrinths  of  the  old  methods,  and 
the  mazes  of  the  old  rule,  in  solving  such  problems.  The  ru?e  and  the  explanation,  under  the  old  sys- 
tem would  cover  several  pages  of  this  work.  By  the  new  method  here  taught,  we  save  the  completion 
ot  the  second  partial  divisor,  and  the  formation  and  completion  of  two  other  divisors,  which  it  is  the 
most  desirable  to  obviate,  because  they  become  very  large  toward  the  end  of  the  extraction,  requiring 
much  time  and  care  in  the  work.  But,  for  the  encouragement  of  the  student,  it  may  be  stated  that  few 
^authors  give  numbers  involving  more  than  four  periods;  and  it  is  also  a  rare  thing  in  applied  mathe- 


MATHEMATICAL   ROOTS   UPROOTED. 


matics  to  find  questions  involving  the  roots  beyond  the  fourth  or  fifth  decimal,  which  are  exceedingly 
easy  by  the  method  here  taught,  but  long  and  tedious  by  the  old  method. 


ADDITIONAL  EXAMPLES  FOR  THE  STUDENT. 


Find  the  cube  root  of  these  numbers,  viz.: 

1.  2176782336. 

2.  87824421 125. 

3.  43132764843. 
Solve,  also,  the  following  : 

7.  ^£TO5 

8. 
9- 


132963364864. 

225199.600704. 

754863.574332608. 


f  173274H 

f2^% 

Note. — All  perfect  cubes  of  only  two  periods  are  to  be  solved  at  sight 
reduced  to  improper  fractions,  or  to  mixed  decimals. 

Find  the  value  of  this  expression,  viz.; 
f  T6"6  --  f  6i-  (4  X  f  Tsl^)  = 
Solution:        162  -f  4  —  (4  X  .8)  = 
256 -r  4  — (   3-2  )  = 

64  —  (  3.2  )  =  60.8.     Atu. 
Find  the  value  of  these,  viz,: 


Mixed  numbers  should  be 


f54.872 
^21.952 


^2326.203125 
^64.964808 


3-     f2357947T^ff 

Answer  to  last :  J^f .     Let  the  student  find  it. 

Note. — If  the  terms  of  a  fraction  are  not  perfect  powers  of  the  indicated  root,  let  them  be  reduced 
to  such  powers,  where  possible,  before  the  extraction  begins. 

The  foregoing  examples  must  suffice  for  illustration  of  the  best  method  of  extracting  the  root  of 
exact  cubes.     The  roots  of  higher  powers  will  be  discussed  in  connection  with  logarithms. 


CUBIC    SURDS. 


We  will  now  present  a  brief,  easy,  and  practical  method  of  treating  imperfect  powers  of  the  third 
degree.  In  advance,  we  state  that  that  method  is,  of  course,  one  of  approximation.  Here  it  is  impos- 
sible to  choose  the  final  figure,  since  there  is  no  definite  final  root  figure  in  surds. 

Let  It  be  required  to  find  the  cube  root,  correct  to  four  decimals,  of  the  fraction 
%.  =  .500000000  I  .7937+ 


ist  divisor 


Finished  divisor 


2d  divisor 


147.81 
189 


1 667 1 
1872.3 


343 


1570.00 
150039 


69610.00  This  answer  is  true  to  the  fourth  place,  inclusive, 

56169  as  verified  by  logarithms.     We  extract  in  the  usual 

way  until  we  obtain  half  the  number  of  root  figures 

13441  desired.     We   then  form   the  second  partial  divisor, 

13 106  and  with  it  obtain  the  other  two  root  figures  by  a 

mode  of   contracted  division,   thus:     Having  found 

335  the  third  figure  of  the  root  and  taken  its  product  with 

the  divisor  from  the  partial  dividend,  use,  instead  of  attaching  other  periods,  the  same  partial  dividend, 
and  drop  one  figure  from  the  right  of  the  divisor  last  found,  and  ascertain  how  often  it  will  go  into  the 
remainder  of  dividend  accruing  from  the  previous  division,  reckoning  in  the  number  that  would  be  to 
carry  from  the  figure  dropped. 


MATHEMATICAL   ROOTS   UPROOTED. 


1st  divisor 


Find  the ^.27  correct  to  four  places. 


108.16 

72, 


"16 

I22&.» 


.270000000  I  .6463-f- 
216 

540.00 


2d  divisor  I228t*         '  Observe  that   16  is  written  in  the  niche  of  the 

other  two  parts  of  the  divisor,  to  save  space. 

Explanation. — 108  is  the  partial  divisor.  (How 
obtained?)  This  divisor  gives  the  next  root  figure. 
1 1 536  is  the  finished  divisor.  (Hov/  obtained  ?)  Ob- 
serve that  the  two  added  parts,  72  and  16,  are  each 
advanced  one  place  to  the  right.  12288  is  the  second 
partial,  or  approximate,  divisor. 

1.  How  is  it  obtained  ? 

2.  In  the  product  of  the  root  figure,  3,  with  the  approximate  divisor,  1228,  account  for  the  figure 
6  in  the  result. 

Obs. — Outside  of  mathematical  astronomy,  where,  in  a  few  instances,  great  precision  is  required,  it 
is  scarcely  necessary  to  approximate  a  root  beyond  four  decimal  figures. 
The  ^.640000000  &c.  =  .8617+ 


7^6q.oo 
73728 

4832 
3686 

1 146 


1st  divisor 


192.36 
144 


20676 


512 

1280.00 
124056 


2d  divisor       2218.8 


Observe  that  36  is  written  in  the  niche  of  the 
other  two  parts  of  the  divisor,  to  save  space.     20676 
is  the  only  finished  divisor  it  is  necessary  to  make. 
1720  After  finding  the  third  root  figure,  we  add  no  other 

periods  to  the  partial  dividend,  but  use  the  same  dividend,  and  use,  as  a  divisor,  the  previous  one  shorn 
of  its  final  figure.  If  we  wish  to  find  other  root  figures,  we  drop  other  figures  from  the  divisor,  one  by 
one,  and  continiie  to  divide  as  in  common  division.  But  it  is  well  to  bear  in  mind  that  if  v/e  wish  to 
obtain  accurately  a  definite  number  of  decimals  in  the  root,  we  must  extract  in  the  ordinary  way,  until 
we  have  obtained  one-half  of  them,  and  then  make  a  partial  divisor  from  the  root  thus  far  found,  and 
use  that  as  an  approximate  divisor,  to  find  the  other  root  figures.  But  this  is  a  very  great  saving,  as  it 
is  the  final  periods  that  are  to  be  dreaded  in  extraction.  Do  not  fail  to  be  familiar  with  the  cubes  i,  8, 
27,  64,  125,  216,  343,  512,  729.     Otherwise  you  cannot  select  at  sight  the  first  root  figure. 

~-  '       ~"       -8735+ 

1st  divisor 


Thef^/^  = 
192.49 
168 


divisor 


20929 


2270.7 


.666,666,666,  &c. 
512 


1546.66 
146503 

81636.66 
68i2i 


135 1 5  See  that  49  is  written  in  the  niche  again,  i'nstead 

1 1353  of  being   written  thus:     192 

168     which  would  occupy 

2162  49.  too  much  space. 

Obs. — In  completing  any  divisor,  we  must  advance  each  part  one  place  to  the  right. 

The  f  .097672831790877  =  .46053+ 
1st  divisor        48.36      I      64 

»72  — 
I     336.72 
5556  33336 


2d  divisor         63480.0 


3368.31 7.90 
3174000 


I 943 I 7 
190440 


3877 


This  is  accurate  as  far  as  extracted ;  and,  ye  , 
there  is  a  necessity  to  frame  only  a  second  approxi- 
mate  divisor.  Referring  to  the  third  root  figure,  we 
see  that  the  divisor  is  not  contained  in  the  partial  divi- 


10 


MATHEMATICAL    ROOTS   UPROOTED. 


dend  shorn  of  the  last  two  figures,  and  we  put  a  nought  in  the  quotient,  and  two  noughts  to  the  right 
of  the  divisor,  because  the  squaring  of  the  root  thus  far  found,  namely  460,  would  give  two  noughts  in 
the  resulting  partial  divisor,  634800.  Attaching  another  period,  we  use  this  divisor  to  find  the  remain- 
ing root  figures. 

Required  the  indicated  roots  of  the  following  surds,  accurate  to  four  decimals,  viz.; 


I. 

f.171467 

3- 

^2.42999 
1^19-44 

4- 

f^.571428 

5- 
6. 

7- 

f  5 
f  4>^ 

f  42f 

8. 

9- 
10. 

f  22.4 
f  3 

1^41-502 
f  II 

1^9^ 
f   9 

f    7 


f48x  4=^-182 

^'300484 

^129.009 


^.6748 
f    6 


.2482 


These  examples  must  suffice  for  illustration  of  the  brevity  secured  by  this  mode  of  approximating 
the  cube  root  of  surds,  by  which  at  least  three-fourths  of  the  work  necessary  under  other  methods  is 
saved  ;  while,  in  perfect  cubes,  nine-tenths  of  the  usual  work  is  obviated. 

HIGHER    ROOTS. 

To  "roots  of  all  powers,"  so  called,  but  a  small  space  can  be  allotted  in  this  brief  work.  Some 
authors,  with  what  seems  to  be  a  strange  love  of  novelty,  rather  than  a  desire  for  utility,  have  made 
quite  an  array  of  numbers  requiring  the  5th,  6th,  7th,  8th,  9th,  lOth,  I2th,  15th,  l8th,  20th;  and  the 
25th  root,  to  be  extracted.  Now,  it  is  needless  to  say  that  no  such  roots  occur  in  nature,  or  in  the 
course  of  applied  mathematics.  It  is  rare,  indeed,  in  applied  mathematics,  that  a  number  or  quantity 
occurs  requiring  a  root  beyond  the  ihh-d  or  fourth.  Then  why  should  such  numbers  encumber  that 
most  practical  of  all  the  mathematical  branches — arithmetic  ?  One  of  the  many  authors  on  arithmet- 
ical science,  whose  works  are  in  extensive  use  in  this  country,  requires  the  20th  root  of  617,  the  15th 
root  of  15,  and  the  25th  root  of  100.  Wherefore?  we  ask  ;  what  the  need  ?  When  will  the  necessity 
for  their  use  arise?  Such  novelties  are  incubuses  on  the  science  of  numbers,  and  ought  to  be  relegated 
to  the  closets  of  defunct  mathematics.  But,  if  mathematicians  must  put  such  impractical  problems  in 
their  books  and  have  them  solved,  let  them  be  solved  by  shorter  and  better  methods  than  those  pre- 
sented in  their  works.  That  briefer  and  better  method  is  by  m.eans  of  logarithms.  Especially  is  this 
true  for  roots  whose  indices  are  not  factorable  into  the  square  and  cube  roots.  Indeed,  even  in  this 
case,  the  logarithmic  method  would  be  far  preferable,  and,  if  once  adopted,  would  supersede  all  other 
methods  for  the  higher  roots.  For,  although  the  8th  root  can  be  taken  by  three  successive  extractions 
of  the  square  root,  the  9th  root  by  two  successive  extractions  of  the  cube  root,  and  the  6th  root  by  the 
cube  root  of  the  square  root,  or  the  square  root  of  the  cube  root,  still  these  roots  can  be  much  more 
easily  and  quickly  taken  by  logarithms.  We  simply  take,  from  a  table  of  logarithms,  the  log.  of  the 
number  whose  root  is  sought ;  divide  this  log.  by  the  index  of  the  root,  and  find,  in  the  same  table,  the 
number  corresponding  to  l^^^uotient,  and  it  will  be  the  required  root. 

Ex. — Find  the  cttb^oot  of  1.577635 — . 

The  log.  of  this  number  is  .19800+ ;  divide  it  by  9,  the  index,  and  the  result  is  .02200-{-,  and  the 
number  in  the  table  corresponding  to  these  figures  is  1. 05 19,  the  required  root.  The  same  result  is  also 
easily  obtained  by  the  abbreviated  method  for  cube  root,  thus: 

1st  root.         2d  root. 
1-577635=  1.1641+  I  I.05-I-.     Ans. 

Divisor       3.3.1         i  i 


Divisor       363.36 
iq8 


Divisor 


383^6^ 
403. 6.  J 


5-77 
331 

2466.35 
229896 

16739 
16147 


3.00 


1. 641. 00  This  is  the  answer  to  the  example  in  the 

1500     ,  work  from  which  it  has  been  drawn  ;  but  it 

will   be   seen    that  the  logarithmic  method 

141  gives  the  answer  more  accurately.    We  have 

simply  taken  the  cube  root  of  the  cube  root 
by  our  method. 


592 
404 


MATHEMATICAL   ROOTS   UPROOTED. 


Find  the  7th  root  of  308.  The  log.  of  308,  page  6  of  the  logarithmic  table,  is  2.48855  ;  divide  by 
the  index  of  the  root,  2.48855  4-  7  =  .35550-]- ;  and  the  number  corresponding  to  this  logarithmic  quo- 
tient is  2.26729  +  ,  the  7th  root  of  308.  All  that  is  necessary  in  order  to  extract,  by  logarithms,  atty 
root,  is  to  look  in  a  table  of  logarithms  for  the  log.  of  the  number  to  be  extracted  ;  divide  this  by  the 
index  of  the  root,  and  find,  in  the  same  table,  the  number  corresponding  to  the  quotient,  and  it  will  l)e 
the  root  sought.  This  method  is  vastly  shorter,  and  only  requires  a  little  knowledge  of  logarithms,  and 
a  little  facility  in  their  use,  to  enable  one  to  evolve,  with  despatch,  all  roots.  But  such  roots  belong 
rather  to  higher  mathematics.  We  would  advise  that,  by  all  means,  all  roots  above  the  third  be  taken 
by  means  of  logarithms.  It  is  but  an  hour's  work  to  teach,  to  anyone  that  can  multiply  and  divide,  the 
use  of  the  table.     The  after  work  is  simply  routine,  and  much  valuable  time  is  saved. 

As  a  matter  of  curiosity,  we  give  below  the  7th  root  of  the  same  number,  as  presented  by  its  author 
in  one  of  the  books  of  the  day.     That  is,  the  7th  root  of  308. 

OPERATION, 
f  338"=  2.59 + 
■1/308=2.04-1- 
2.59-1-2.04  =  4.63 
4.63 -=- 2  =  2.31 -|-     assumed  root. 
2.316  =  151.93 
308 -M51. 93  =  2-0272 -h 
2.31X6-1-2.0272=15.8872 
15.8872  -f  7  =  2.2696     1st  approximation. 
2.2696®  =  136.6748 
308-^136.6748  =  2.253452-}- 

2  2606  V  6 -^  2  2^'i±<2  —It;  87io-;2  ^'^  ^^^'^  reached  the  second  approximation,    Con- 

2.2090  X  0  -h  2.253452  -15.671052  g^j.^^^  thought !     And  these  are  only  indicated  results, 

15.871052^7  =  2.267293     2d  approx.  none  of   the  multiplications,  divisions,  etc.,  being  car- 

ried out.  Now,  if  this  belongs  anywhere,  and  there  is  doubt  of  its  having  a  place  in  applied  mathe- 
matics, it  belongs  to  higher  mathematics.  Such  skirmishing  in  figures  is  calculated  to  keep  one  hum- 
ble, by  giving  him  a  modest  estimate  of  his  attainments  in  evolution. 

EVOLUTION  BY  LOGARITHMS. 

Required  the  25th  root  of  100.  The  log.  of  100  =  2.000000.  2.000000  -f-  25  =  .080000,  and  the 
number  corresponding  is  1.202266  +  ,  which  is  the  root  sought.  The  solution  of  the  same  example,  as 
given  by  a  standard  author,  is  as  follows  : 

Now,   as   the    The  y/ 100  =10 
25th   root  must    / 100  =  y  10  =3.1622 
be  less  than  the    y  100  =  /3.i622^  1.7782 
24th  root,  let  us    |^'ioo=  ^1.7782=1.2115 
take  1.2  =  the  assumed  root. 

1.224  =  79.49684-]- 
100 -^  79,49684=  1.25792 -f 
1.2  X  24  -f  1.25792  =  30.05792 

30.05792  -T-  25  =  1. 2023168     1st  approx. 
1.2023x682*  =83.2677184 
100-^83.2677184=  1.2009492-f- 
1.2023168  X  24-j-  1,2009492^=30.0565524 

30.0565524  -^  25  =1.202262-]-  2d  approx. 
We  breathe  a  sigh  of  relief.  Of  course,  not  much  space  can  be  devoted,  ii^  this  small  work,  to  such 
solutions.  It  is  only  to  show  the  difficulty  of  the  subject  under  the  old  methods,  in  contrast  with  the 
brevity  and  facility  of  the  new,  that  we  allow  a  little  space  for  some  solutions  under  existing  methods. 
Find,  by  the  logarithmic  method,  the  6th  root  of  25632972850442049.  The  log.  of  this  large  num- 
ber is  16.408800.  Dividing  by  6,  we  get  2.734800.  The  number  found  in  the  table,  corresponding  to 
this  quotient,  is  543,  the  sixth  root  of  the  above  number.  The  foregoing  number,  treated  by  the  new 
method,  gives,  for  the  square  root,  160103007,  and  the 
^160103007  =  543 

125  Explanation. — With  the  approximate   divisor,  75,  find   the  second 

figure  of  the  root,  4,  and  choose  the  last.     Why  is  it  3  ?     Thus,  the 

75       351-03  work  of  the  heretofore  difficult  cube  root  is  almost  annihilated. 


12  MATHEMATICAL   ROOTS   UPROOTED. 

What  is  the  20th  root  of  617  ?  The  log,  of  617  is  2.790285.  Dividing  by  20,  we  have  .139514, 
and  the  number  corresponding  is  1,378841,  the  ans. 

Find  the  5th  root  of  5,  The  log.  of  5  is  .69897.  Divide  by  5,  and  get  .13979,  and  the  number  cor- 
responding is  1.37973,  the  5th  root  of  5. 

The  5th  root  of  120  is:  Log.  120  =  2^piS=  .41583 -(- ;  and  the  number  corresponding  is  2.605174:5 
the  root  wanted.  ^  ^ 

Let  the  student  find,  by  logarithms  (see  explanation  of  use  of  the  table,  pp.  -62  and^,  etc.,)  the 
roots  of  the  following  numbers,  viz.: 


1.  The  8th  root  of  109951 1627776, 

2.  The  I2th  root  of  16.3939. 

3.  The  i8th  root  of   104.9617, 


4,  The   7th  root  of  1.95678. 

5,  The  loth  root  of  743044. 

6,  The  3rd  root  of  4330747. ' 


Find,  also,  by  logarithms,  or  by  the  abbreviated  method,  at  your  option,  the  cube  root  of  the  fol- 
lowing numbers,  viz.: 

7.  7023 1 089 1 84307 2.  I  9.     10964743589696. 

8.  744935304423023.  I  10,       1881365963625. 
Required  the  solution  of  these  examples  :  '^"^' 

1.  If  a  ball  10  inches  in  diameter  weighs  125  lbs,,  what  is  the  diameter  of  a  ball  that  weighs 
216  lbs.?  

Solution.— f  \2i^  :  ^216  ::  10 :  ans=  ^f||  X  lO  ==  12, 
5:6::  10:  ans,  Ans.  12  inches. 

2.  How  many  balls  %  inch  in  diameter  will  be  required  to  make  a  ball  i  inch  in  diameter  ?     Ans. 
64  balls. 

3.  Suppose  the  diameter  of  the  earth  to  be  7912  miles,  and   that  it  takes  1404928  bodies  of  the 
size  of  the  earth  to  make  one  as  large  as  the  sun,  what  is  the  diameter  of  the  sun  ? 

^i3o|2lsx  7912=  112  X  7912  =  886144  miles.     Ans. 

4.  A  bin  is  8  feet  long,  4  feet  wide,  and  2  feet  deep  ;  what  is  the  linear  edge  of  a  cubical  box  that 
will  hold  the  same  quantity  of  grain  ? 

t^S  X  4  X  2  =  f8x8  =  2X2  =  4  feet.     Ans. 
Let  the  curt  processes  be  used.     Extract  the  factors  of  products  in  preference  to   taking  the  roots 
of  the  products. 

5.  If  a  stack  of  hay  24  feet  high  weighs  27  tons,  what  is  the  hight  of  a  stack  weighing  8  tons? 

if  Ty  X  24  =  2^  X  24=  16  feet.     Ans. 

6.  If  a  bell  4  inches  high,  3  inches  in  diameter,  and  "%  of  an  inch  thick,  weighs  i  pound,  what  are 
the  dimensions  of  a  similar  bell  weighing  27  pounds  ? 

Ans.     12  inches  high,  9  inches  diameter,  and  ^  of  an  inch  thick. 

7.  If  a  loaf  of  sugar  10  inches  high  weighs  8  pounds,  what  is  the  hight  of  a  similar  loaf  weighing 
I  pound  ? 

fyiY,  10  =  j^  X  10=  5  inches.     Ans. 

8.  There  is  a  bin  32  ft.  long,  16  ft.  wide,  and  8  ft.   deep;  what  must  the  side  of  a  cubic  bin  be 
that  shall  contain  the  same  quantity  ? 

^32  X  16X8=  1^64X64  =  4X4=  16  ft.     Ans. 

9.  What  must  be  the  side  of  a  cubic  bin  that  shall  hold  350  bushels  of  grain  ? 

SOLUTION. 
2150,4X350  =  752640  I  90.96+ =7  ft.  6,96  in. 
Divisor         24300.81     729  Ans. 

2430 

236,400.00 
22089429 


Fin.  div.      245438. 


1550571  We  use  the  finished  divisor,  shorn  of  the  final  fig- 

,    .  1472629  ure,   as  an  approximate  divisor  to  obtain  the   fourth 

figure  of  the  root.     For  practical  purposes,  the  above 

77942  is  a  close  approximation. 

10.  If  a  sphere  of  gold  i  inch  in  diameter  is  worth  $100,  what  is  the  diameter  of  a  sphere  that  is 
worth  $6400  ? 

iff^Q  =  ^64  X  I  =  4  inches.     Ans. 

11.  The  cubic  metre  is  61026.048  cubic  inches  ;  what  is  the  linear  metre  ? 

Ans.     39.37  inches.     Find  it  by  approximation. 
We  give  one  more  illustration,  each,  of  the  new  method  of  taking  the  cube  root  of  perfect  and  im- 
perfect third  powers : 


MATHEMATICAL    ROOTS    UPROOTED. 


1st  divisor 


2d  divisor 


The  ^146113369163  =  5267.     Ans. 


75 
30-4 

7804 


125 

211.13 

15608 


55053-69  There  is  no  necessity  of  the  last  multiplication,  6  times 

48672  the  second  divisor.     Satisfy  yourself  that  the  remainder  will  be 

less  than  the  divisor,  and  then  choose  the  last  root  figure,  thus 

6381  saving  a  vast  amount  of  work.    Why  is  the  final  root  figure  7? 

What  is  the  value  of  1.05I  to  6  decimals? 

The  log  of  1.05  is  .021189.  Divide  this  by  3,  and  multiply  the  result  by  5,  and  we  get  .035315 ;  the 
number  corresponding  is  i. 084715 +  ,  the  answer.  To  solve  the  above  example  in  the  old  way,  will  re- 
quire about  30  minutes;  by  the  logarithmic  method,  2  or  3  minutes. 

The  ^1.1810108914205625,  by  the  approximate  method,  accurate  to  6  decimals,  is  1.0570234;. 
Find  it.     ^   ^^  ,  p  ^    ^  ,-^^  T   v^  A  i  V  5     ;  .:,  ;    ].  OSI^O  ^  :i  ,  ^>  /  s^  %   -■  ^  >      ^  :"■ 

We  have  presented  an  unusually  large  number  of  solutions,  in  order  that  the  cube  root  method 
herein  set  forth  may  be  clearly  apprehended  by  all.  For,  knowing  its  advantage  in  brevity  and  sim- 
plicity, and,  consequently,  its  economy  of  time  and  space,  we  are  thoroughly  convinced  that,  if  once 
adopted,  it  will  be  abandoned  for  no  other.  Let  it  be  remembered,  that  if,  in  approximating  the  cube 
root  of  a  number,  it  is  desirable  to  extend  the  answer  to  a  given  number  of  decimal  figures,  one-half  of 
all  the  root  figures  must  be  obtained  by  extraction  in  the  usual  way.  The  other  half  may  be  obtained 
by  contracted  division.  For  instance,  in  example  9  of  the  12th  page,  90.9  is  obtained  by  extraction, 
and  625  is  obtained  by  contracted  division.  If  it  be  asked  how  we  determine  when  a  number  is  a  per- 
fect cube,  and  when  it  is  a  cubic  surd,  we  answer  that,  in  actual  business,  in  applied  mathematics,  this 
fact  is  always  known  when  the  problem  occurs  in  the  course  of  our  work.  Perfect  cubes  are  in  the 
minority  in  the  course  of  mathematics.  The  small  number  of  problems  given  under  the  head  of  evolu- 
tion by  logarithms,  will  be  sufficient  to  illustrate  the  subject.  The  student  may  work  any,  or  all,  of  the 
others  by  logarithms,  if  he  chooses.  For  this  purpose,  a  table  of  logarithms  is  appended,  calculated  as 
accurately  as  possible  to  five  decimal  places.  The  table  is  extensive  enough  to  enable  us  to  find  the 
roots  of  all  numbers  correct  to  five  decimals.  An  explanation  of  the  use  of  the  table  is  also  appended. 
Should  anyone  desire  further  aid  in  the  matter  of  a  knowledge  and  ready  application  of  logarithms  in 
the  extraction  of  roots,  the  author  will  take  pleasure  in  rendering  all  the  assistance  in  his  power.  In 
conclusion,  if  any  have  their  pet  theories,  methods,  or  processes,  in  cube  root,  to  which  they  cleave 
with  such  a  blind  adherence  that  they  cannot,  or  will  not,  see  merit  in  anything  else,  this  book  is  not 
made  for  them.  If  any  are  moved  by  prejudice,or  jealousy,  or  envy  at  my  good  fortune,  or  by  a  spirit 
of  criticism,  or  are  unduly  inflated  with  the  importance  of  their  own  knowledge  of  the  subject,  through 
the  belief  that  nothing  new  can  be  presented  in  evolution,  the  book  is  not  written  for  any  of  these  classes. 

All  true  science  consists,  not  in  the  discovery  of  any  new  truth, but  in  the  right  application  of  ex- 
isting truth,  so  as  to  render  it  subservient,  in  the  highest  degree,  to  the  interests  and  to  the  pleasure  of 
mankind.  If  *'  brevity  is  the  soul  of  wit,"  it  is  no  less  the  key  to  successful  business.  The  large  cur- 
tailment of  the  amount  of  work  done  in  book-keeping  in  the  last  few  years,  is  only  in  harmony  with  the 
spirit  of  the  age,  and,  reinforces  the  sentiment  of  "short  profits  and  quick  sales."  So  must  our 
methods  and  processes  in  education  be  constantly  improved  and  refined,  so  as  to  be  the  most  highly 
contributory  to  the  important  interests  of  business  and  of  society. 


14  MATHEMATICAL    ROOTS    UPROOTED. 


Simple  Interest. 


Owing  td  the  universal  application  and  great  practicality  of  this  subject,  we  have  thought  proper'  to 
give  it  a  place  in  this  work,  and  a  treatment  that,  for  brevity,  utility,  and  simplicity,  is  in  keeping  with 
the  constant  drafts  made  upon  it  by  all  classes  of  men — those  of  inferior,  as  well  as  these  of  superior, 
attainments.  The  subject  is  what  the  name  signifies,  but  is  made  rather  complex  by  some  authors  and 
teachers,  owing  to  the  multiplicity  of  rules  and  tedious  methods  of  treatment  used  by  them.  In  simple 
interest,  there  is  scarcely  a  necessity  for  more  than  one  uniform  rule,  whatever  be  the  rate  or  the  time. 

Let  us  take  some  examples,  by  way  of  illustration : 

What  is  the  interest  of  $450.87  for  i  yr.  7  m.  and  9  da.,  at  6  per  cent.? 

Operation. —  450.87 

.0965 


.005  225435 

450.87     19.3    >&6.  270522 

X — X —  =  405783 

I  ^•i,       I  9  days  is  .3  of  a  month,  and  the  process 

Ans.     43.508,955         is  simply  one  of  cancellation. 

Find  the  interest  of  $125.16  for  i  yr.  li  m.  25  da.,  at  6  per  cent.? 
Process. —  1.4298 

10.43 
10.43 


'■^    23.83+.06  42894 

-X  —  X — ==  57192 


I     -13.    I         14298 


One  year  and  Ii  months  are  23  months,  and 
An%.     14.91        25   d.   are  .83+  of  a  mo.     So  that   the  time  is 
23-834-  mos.,  or  I2ths  of  years,  at  the  given  rate  per  year.     Let  the  cancellations,  and  all  the  multi- 
plications possible,  be  done  mentally. 

Find  the  interest  of  $1500.60  for  2  yr.  4  mo.,  at  6^  per  cent.? 

125.05 

1-75 

125.05  

>5aQ.6o    28    .06  >(  62525 

X-X =  87535 

I       ^ta.,     I  12505 

We  take  6  times  28,  plus  the  %  of  28, 

Ans.     $218.8375        mentally,  which  gives  1.75. 

Find  the  amount  of  $3050  for  4  yr.  8  m.,  at  51^  per  cent.? 

i.245>^ 
3050 

14     .0175  

3050  ^   ^e5^  62250 

X-X =  3735 

I     -«        I  

-3-  Ans.    $3797.250  In  making  multiplications  mentally,  after 

the  cancellations  have  been  made,  let  the  smallest  numbers  be  so  multiplied.  Thus  14  times  .0175,  ^"^ 
then  add  i  to  the  result,  to  get  the  amount  of  $1  for  the  lime,  at  the  given  rate.  This  result  is  then 
multiplied  by  the  principal. 


MATHEMATICAL   ROOTS   UPROOTED. 


15 


Required  the  interest  of  $250  for  i  yr.  10  m.  15  da.,  6  per  cent. 
125                .01  .225 

--^^  22.5   rD6-  125 

I         "T^  I  II25 

•'^  2700 


Ans.     $28,125 

If  the  amount  had  been  required, 

we 

should  have 

proceeded  thus : 

.005 

1. 1125 

250    22.5     ^36 

250 

I      -t^        I 

556250 

4ns, 

22250 

$278.1250 

After  the  mental  multiplication  of  the 
time  and  rate,  add  one  to  the  result,  be- 
fore the  final  multiplication  by  the  prin- 
ciple. 


Find  the  interest  of  $51.10  for  10  m.'and  3  da.,  4  per  cent. 
.01  51.10 

51.10     lo.i    Tc^.  .101 

I         r^       I  511 

3  5" 


3|5-i6ii 

Ans,     1.72 

It  may  also  be  done  thus  : 

===== 

17-033+                 -01 

17.033 

-3tTT&     10. 1      Aft^ 

.101 

X X  — = 

I         ^s^       I 

17033 

^ 

17033 

As  before     $1.720333 


What  is  the  interest  of  $175.40  for  15  m.  8  da.,  10  per  cent.? 
,     1.272+                           175.40 
175.40   >5,.i66+    .10                 .1272 
X X  — =  

I  ^Ki  I  3508 


12278 
21048 

Ans.     $22.3108 


The  multiplications  are  made  mentally, 
except  one. 


Required  the  amount  of  $1500  for  6  m.  24  da.,  JJ^  per  cent. 
7     -025 

1.0425  =amt.  of  $1. 
1500 


1500  ~'&r5    >f5, 

I          T-3  I 

:3 


$1563.7500     Ans. 


Required  the  amount  of 


1.25  for  I  yr.  5  m.  10  da.,  6}{  per  cent. 
8664 
361 


1.444+ 
84.25    >>^+     .o6X_ 

I  >2.  I 


1.09025 
84.25 

545125 
218050 
436100 
872200 


Ans.  .   $91.85 


l6  MATHEMATICAL   ROOTS   UPROOTED. 

After  first  multiplication,  add  I  to  the  result,  before  the  formal  multiplication. 
Find  the  interest  of  $112.50  for  3  m.  I  da.  <)%  per  cent.? 
.2527+  2274  112.50 

112.50    ^TS954-    .09^        126  .02400 

I  1^  I  4500 

2250 


Ans.    $2.70 
In  multiplying  the  time  and  rate  together,  allow  for  the  number  of  units  that  would  be  carried 
from  9  times  7. 

What  is  the  interest  of  $408  for  20  da.,  6  per  cent.? 

.333        .01  408 

408    ":64i44-    7o§-.  .00333 


1224 
1224 
1224 


1. 35864=$!. 36,     Ans. 
Always  divide  the  days  by  3,  since  every  three  days  is  tV  of  a  month,  thus  reducing  the  days  to 
decimals  of  a  month.    More  than  5  mills  should  be  called  a  cent.    Less  than  5  mills  should  be  disregarded. 

Required  the  interest  of  $50  for  i  yr.,  3  m.,  27  da.,  3 >^  per  cent. 
25       5-3     .05  1-25 

-5€-  -f^rf-    .-K-^  5.3 

I       -t5-     -3-  375 

-6-  625 

^    ^  3  16^6^ 

Ans.     $2.2o8>^ 
Find  the  interest  of  $384.50  for  2  yr.,  8  m.,  4  da.,  8  per  cent. 
10.71         .02  384.50 

384.50  -:5^~iad-    -"^^        -2142 

I         -«-  I  7690 

3-  15380 

3845 
7690 


5$82.35990=$82.36.    Ans. 
Solve  this  by  other  methods  and  see  if  you  get  a  different  answer. 

Required  the  amount  of  $275  in  4  m.,  25  da.,  7  per  cent. 
.4027+  1. 02818 

275    >>S5i+-      -07  275 

— X X  — =       

I         -tt-  I  514090 

719726 
205636 


$282.7495o=-$282.75.     Ans.    ^^    ^  ^  ,     .v    1    * 

After  multiplying  .4027  by  .07,  we  add  i  to  the  result,  for  the  amount  of  $1,  before  we  make  the  last 
multiplication.     Multiply  .402  by  .07,  but  carry  from  7  times  7. 
Interest  of  $318.29  for  9  m.,  10  da.,  "]}(  per  cent.? 
•777+  5443        318.29 

318.29  -9:395;^     .07X       194        .05637 

X X = —       

I     -in-  I         222803 

95487 
190974 
159145 


Ans.     17.94.20 


MATHEMATICAL    ROOTS    UPROOTED. 


17 


Required  the  interest  of  $4684.68  for  11  da.,  12;^  per  cent. 

390-39 
.04582 


390.39     .183+ 
,+§S:;:cS.  -rS^-l-     .25  — -— 
X X =  78078 

I  -K-  -2-  3I23I2 

I95I95 
I56I56 


$17.8876  =  $17.89.     Arts. 


In  making  the  mental  multiplication  by 
25,  allow  for  the  number  that  would  be  to 
carry,  had  the  decimal  .183+  been  extended 
one  figure  further. 


Find  the  interest  of  $127.36  for  I  yr.  6  m.  21  da.,  4^  per  cent. 

5-306+ 
5.306+  .  1.683^ 

-is?.^^-    18.7     .09  15918 

X X =  42448 

I  -r^     -2-  31836 

5306 


Ans.     $8.93.  $8.929998 

Find  the  amount  of  $723.60  for  2  yr.  3  m.  18  da.,  5^  per  cent, 

2-3 
723.60  ^Jfr^.    .23       .529 

X — X—  =  — =  .13225 

I  -f~      4  4 

Add  I 


1. 13225 
723.6 

679350 
339675 
22645b 
792575  After  cancelling  and  multiplying  the  expres- 

sions  of  time  and  rate,  we  add  i  to  the  product, 

$819,296  Ans.  to  get  the  amt.  of  $1.  This  saves  time  and  one 
operation  in  the  work.  Now,  if  the  work  is  short  with  these  peculiar  and  mixed  rates,  it  is  much  mo-e 
so  with  all  ordinary  rates.     We  will  take  only  two  or  three  illustrations  : 

Find  the  interest  of  $780.26  for  90  da.,  without  grace,  at  1%.  per  cent,  per  month. 
Operation, —  195.065 

195.065  .15 

;So.x6 — 3^  .15  

— X— X— ==  975325 

I  -rt-    I  195065 

_4_  

Ans.     $29.26975 
What  is  the  interest  of  $845  for  i  yn  10  m.  6  da.,  at  I  per  cent  a  month? 

845 
.01  .222 

845     2^.2     7t±-  

X X =  1690 

I       -1-3-        I  1690 

1690 

Multiplying  by  .01  simply  throws  the  point 

Ans.     $187,590     two  placesfurther  to  the  left  on  the  multiplicand. 
Find  the  interest  of  1040  for  i  yr.  9  m.  9  da.  at  8  per  cent. 

.142 
7.1     .02  1040 

1040  -2*t5  "naS^  

X X- 5680 

I         nr        I  142 

-3-  

Ans.     $147,680         This  method  is  equally  expeditious  in   reck- 
oning up  notes  whereon  partial  payments  have  been  made.     Indeed,  there  is  no  department  of  interest 


i8 


MATHEMATICAL   ROOTS   UPROOTED. 


where  it  may  not  be  used  with  greater  facility,  and  with  much  less  work,  than  any  other  process.  We 
have  chosen  to  call  it  the  Cancellation  Method.  The  advantage  lies  in  its  simplicity,  brevity,  and  uni- 
formity of  treatment,  there  being  but  one  process  for  all  problems,  whatever  the  rate,  time  or  other 
conditions.  And  surely  this,  of  itself,  is  a  great  saving,  to  both  teacher  and  pupil,  of  much  labor  and 
taxing  of  memory,  under  the  numerous  methods  and  rules  of  interest  laid  down  in  the  books.  The 
plan  here  presented  is  strictly  mathematical,  depending  on  the  principle,  that  the  annual  rate  on  a  dol- 
lar, multiplied  by  the  number  of  dollars  in  the  principal,  is  equal  to  the  interest.  The  time  is  reduced 
to  months  and  decimals  of  a  month;  and,  then,  the  expression  for  months  is  divided  by  12,  thus  ex- 
pressing the  time  in  years.  Each  example  given  in  the  foregoing  pages,  is  simply  a  grouping  of  the 
sum  at  interest,  the  years,  and  the  rate.  Cancellation  naturally  follows.  We  might  have  reduced  the 
time  to  days,  dividing  the  number  ot  days  by  360,  thus  making  it  express  years.  For  instance,  2  yr.  4 
m.  20  da.  is  720  da.  -f-  120  da.  -|-20  da.  =  860  da.,  or  ||g  years.  But  this  is  considerably  longer,  requir- 
ing more  work  every  way.  The  briefer  the  method  of  reckoning  interest,  the  less  liability  to  mistakes. 
The  one  herein  set  forth,'  takes  the  happy  mean  in  all  particulars. 


EXAMPLES  FOR  THE  STUDENT. 


2. 
3- 
4- 
5- 
6. 

7- 
8. 
9- 
10. 


Let  it  be  required  to  find  the  interest  on  the  following,  viz.: 
$300  for  2  yr.  7  m.  24  da.,  at  6  per  cent. 
$700  for  I  yr.  9  m.  12  da.,  at  6  per  cent. 
$400  for  2  yr.  6  m.  15  da.,  at  6  per  cent. 
$350  for  3  yr.  8  m.  24  da.,  at  6  per  cent. 
$450.87  for  I  yr.  7  m.  9  da.,  at  7  per  cent. 
$375.50  for  2  yr.  i  m.  8  da.,  at  7  per  cent. 
$125.16  for  I  yr.  ii  m,  25  da.,  at  7  per  cent. 
$658.25  for  I  yr.  2  m.  13  da.,  at  7  per  cent.      1 
$187.44  for  I  yr.  10  m.  24  da.,  at  7^,7  per  cent.  [-Find  the  amount. 
$444.84  for  I  yr.  i  m.  16  da.,  at  5  per  cent.       j 
Also,  reckon  up  the  following  promissory  notes,  on  which  indorsements  have  been  made,  viz.: 
$167.42.  Selma,  Apr.  15,  1882. 

1.  For  value  received,  I  promise  to  pay  Judge  Fowler,  or  order,  one  hundred  sixty-seven  and  -^ 
dollars,  in  6  months  from  date  with  interest. 

Tom  Scroggins. 
Indorsements  : 

May  21,  1883,  $42.18;  July  17,   1884,  $6.25;  Sept.  9,   1884,  $48.16;  Jan.  27,  1885,  $27.47.     What 
was  due  Apr.  15,  1886?  ^ns.     $72,277. 

$472.76.  Selma,  June  4,  1884. 

2.  For  value  received  of  Arrents  &  Longacre,  I  promise  to  pay  them,  or  their  order,  four  hundred 
seventy-two  and  ^^%  dollars,  in  6  months  from  date,  with  interest  at  7  per  cent,  afterward. 

Jno.  Grubs. 
Indorsements  : 

Apr.  10,  1885,  $125,843  ;  Nov.  28,  1885,  $133,724;  Apr.  15,  1886,  $223,081.     What  will   due  Nov. 
13,  1886?  ylns.     $24.95. 

Let  these  be  done  strictly  by  the  abbreviated  process — it  saves  half  the  usual  work.     These  exam- 
ples must  suffice  for  our  book.     The  student  will  find  abundant  material  for  practice  from  other  sources. 
We  will  present  the  solution  of  the  last  promisory  note,  to  show  the  plan  by  the  cancellation  process. 
W^e  first  write  the  dates  in  succession,  thus ; 
1886,  II,  13  =6  m.  28  da. 
1886,    4,  15  =  4  m.  17  da. 
1885,  II,  28  =  7  m.  18  da. 

4,  10  =  4  m.    6  da.  JV.  B. — The  four  cancellations  and  multiplications"  following,   present 

12,    4=  the  entire  work,  and  not  simply  the  indicated vioxY'. 


1885, 


472.76 
1.0245 


236380 
189104 

94552 
47276 


$484,342 
Payment — 125.843 


$358-499 


.63+ 
358.50    .i^  .07 
X X 


358-50 
1.0443 

10755 
14340 
14340 
3585 


$374,381 
Payment— 133.724 


$240,657 


MATHEMATICAL   ROOTS   UPROOTED. 


19 


1.026635 
240.657 


•3805 

240.657  4.566  U 

X X— 

I     -^-^-        I 


07 


7186445 

5133175 
6159810 
4106540 
2053270 


$247.06689 
Payment— ^223 .08 1 
$23.99 


•574- 
23-99    -^^^^f     -07 

I  -irt-         I 


•57  X. 07+ I  =  1.0399 


$24,947  =  $24.95. 
Ans. 
ramt.  of  $1. 


In  partial  payments,  most  authors,  in  illustrating  their  methods,  give  simply  a  brief  of  results,  as  if 
they  would  make  the  work  appear  short.  After  cancellation  and  multiplication  of  the  expressions  lor 
time  and  rate,  let  i  be  added  to  the  result,  for  the  amt.  of  $1,  before  the  mechanical  multiplication  is 
made.  See  above.  Observe  the  manner  of  writing  the  dates,  all  at  once,  and  all  in  one  group,  from 
the  latest  to  the  earliest,  and  then  the  consecutive  subtraction  of  them,  thus  giving  the  several  periods 
of  time  at  once,  before  the  cancellation  processes  are  commenced.  This  is  a  great  saving  of  time,  and 
promotes  simplicity.  And  here  it  may  be  stated  that,  with  respect  to  the  subject  of  interest,  the  author 
does  not  so  much  claim  to  have  discovered  new  truth,  as  he  does  a  new  and  right  application  of  it. 
Much  of  scientific  truth  is  as  good  as  lost,  through  the  circuitous,  obscure  processes  under  which  it  is 
presented. 


Required  the  interest  on  the  following,  viz.: 

1.  $1284.60  for  5  m.  12  da.,  at  ^  per  cent,  a  month. 

2.  $621.09  for  7  m.  16  da.,  at  ^  per  cent,  a  month. 

3.  $818.26  for  9  m.  3  da.,  at  \  per  cent,  a  month. 

4.  $220.38  for  2  m.  21  da.,  at  lOj^  per  cent,  per  annum. 

5.  $62.96  for  I  yr.  8  m.  23  da.,  at  11  per  cent  per  annum. 

62.96 
1-73,  .1903 


62.96  —20^76-+- 


-X X = 


1888 
56664 
6296 


$11.98.     Ans.  to  last. 


$614.42. 

For  value  received,  I  promise  to  pay  D 
on  demand,  with  interest  at  6^  per  cent. 
Indorsements  : 

May  15,  1886,  $169.30;  June   10,  1886,  $88.40;  Sept.  I 
on  this  note  Nov.  20,  1886? 


Selma,  Cal.,  May  i,  1886. 
Wagner,  or   order,  six  hundred,  fourteen  and  -^  dollars, 

John  Davis. 


1886,  $325.80.     How  much  will  be  due 


Write  the  dates  in  a  group,  as  above  ;  begin  with  the  date  of  giving  the  note,  and  subtract  each 
from  the  next  succeeding,  as  5  m.  i  da.  from  5  m.  15  da.,  always  making  the  subtractions  mentally,  and 
writing  the  payments  opposite  the  intervening  times.  Find  the  amount  of  the  original  principal  for  the 
first  term  of  time,  and  of  each  succeeding  principal  for  its  term  of  time,  subtracting  each  payment,  in 
order,  from  the  corresponding  amount,  till  you  come  to  the  maturity  of  the  note.  Then  find  the 
amount  of  the  last  principal  for  the  corresponding  term  of  time,  and  it  will  be  the  balance. 
Solution  of  the  last  example :  614.42 

1.00262    amt.  of  $1. 


1886, 

II, 

20  =  2  m. 

2  da. 

PAYMENTS 

1886, 

9, 

18  =  3  m. 

8  da. 

1886, 

6, 

10  =  0  m. 

25  da. 

$325.80. 

IS86, 

S, 

15  =  om. 

14  da. 

88.40. 

1886, 

5, 

I 

169.30. 

.0291 

-.1165-  .09 

614.42  ~:^f6B-     .2f- 

X X  — 

I        -12-     -4- 


122884 
368652 
122884 
61442 


$616.03 
169.30 


$446.73    2d  prin. 


20  MATHEMATICAL   ROOTS    UPROOTED, 


1.00468     amt.  of  $1. 
44673 


20S      .02^  357384 

446.73    .-853.  -r^T^  268038 

X X 178692 

I         -«-    S^  44673 

$448.82 
88.40 


$360.42    3d  prin. 

1.01836     amt.  of  $1. 
360.42 


.204     .09  203672 

360.42  -3-^6^ :  >^  407344 

X X — =  611016 

I          -13—    _4_  305508 


After  multiplying  .204 


$367,037 
325.80 


by  .09,  prefix  I  to  the  result.      $41.24    4tli  prin. 

I.01162     amt.  of  $1. 
41.24 


404648 
202324 
101 162 
404648 


$41,719.  Ans.  $41.72    Ba/. 

The  above  is  the  entire  work.  It  is  comparatively  short,  and  is  quickly  done.  Let  the  student  use 
the  stereotyped  methods  in  use,  and  he  will  at  once  see  a  vast  difference  in  favor  of  this  curt  cancella- 
tion process,  uniform  for  a/l  rates,  times,  and  conditions,  and  equally  easy  for  all  questions  in  interest. 
For  these  and  other  reasons,  this  method,  once  adopted,  will  be  used  in  preference  to  any  other. 

In  reckoning  up  the  balance  due  on  promissory  notes  whereon  partial  payments  have  been  made, 
let  the  cancellations  be  so  managed  that  the  uncancelled  factors  may,  as  far  as  possible,  be  multiplied 
iogQihev  mentally ;  or,  at  least,  maybe  reduced  to  one  formal  multiplication.  The  advantage  of  the 
cancellation  process  may  be  seen  in  the  following  problem : 

Find  the  amount  of  $235.18  for  2  yr.  8  m.  and  12  da.,  at  5>^  per  cent. 

$235.18 
.9  I. 144 

•w>8.  

235.18    :yM.    -16  94072 

X X =  94072 

I  Hri-    -5-.  258698 


$269.04592  =  $269,046.     Ans. 

Observe  that,  after  multiplying  together  the  expressions  of  time  and  rate,  .g  and  .16,  we  add  I  to 
the  result,  which  makes  the  amt.  of  $1.  This  multiplied  into  the  principal  gives  the  amt.  of  the  debt. 
Hence,  in  the  cancellations,  it  is  not  proper  to  cut  down  the  principal,  when  the  amount  is  to  be  found. 
If  only  the  interest  is  required,  factors  may  be  stricken  from  any  of  the  three  parts. 

Those  who  have  not  tried  this  method  cannot  realize  how  easily  these  factors  (principal,  time  in 
years,  and  rate)  can  be  thrown  together,  and  cut  down  to  the  answer,  with  a  very  small  amount  of  fig- 
uring, whatever  be  the  nature  of  the  parts.  The  years  and  months  are  reduced  to  months,  simply  by 
inspection,  without  a  mental  effort ;  the  days  are  reduced  to  the  decimal  of  a  month,  by  dividing  them 
by  3  ;  and  under  this  mixed  expression  we  place  12,  thus  expressing  the  whole  time  in  years. 

We  have  little  patience  with  sticklers  for  analysis  in  everything.  It  is  very  essential  in  some  depart- 
ments of  arithmetical  science,  but  is  utterly  useless  in  many  of  its  most  practical  subjects.  As  an  inci- 
dental feature,  and  as  conducing  to  thoroughness  in  the  fundamental  principles  of  a  mathematical  edu- 
cation, analysis,  in  a  few  subjects  in  arithmetic,  should  be  thoroughly  taught  to  the  young,  but  beyond 
this  it  is  useless.  In  interest  it  is  of  no  value.  The  banker,  the  lawyer,  the  real  estete  man,  and  all 
other  practical  people,  have  no  use  for  analysis  in  any  of  the  several  departments  of  interest,  but  must 
have  the  most  direct,  curt  processes  for  all  problems  arising  under  this  broad  department  in  the  science 


MATHEMATICAL  ROOTS  UPROOTED.  21 

of  numbers.  Teachers  should  not  lose  sight  of  the  fact  that,  in  our  practical  civilization,  we  are,  in 
many  things,  reducing  more  and  more  to  practicality.  Hence,  as  business  increases  and  ramifies  into 
various  new  departments,  thus  multiplying  our  cares  and  duties,  we  must  seek  to  do  our  work  with  the 
utmost  despatch  consistent  with  accuracy.  The  cancellation  process  in  interest  meets  the  demands  of 
business,  on  the  subject  to  which  it  belongs.  It  supersedes  the  necessity  and  the  utility  of  any  6  per 
cent,  method,  or  12  per  cent,  method,  or  i  per  cent,  method,  or  10  percent  method,  or  any  other 
specific  method.    It  secures  uniformity,  simplicity,  brevity,  accuracy, 


FINIS. 


22  LOGARITHMS. 


Logarithms. 

A  Logarithm  is  simply  an  exponent  of  a  power.  The  logarithm  of  a  number  is  the  exponent  of  the 
power  to  which  the  base  of  the  number  must  be  raised,  to  produce  the  number.  Thus,  in  the  following 
equations : 

5"  =  1     and  lo<*  =  I 

51  =  5  loi  =  10 

52  =  25  lo*  =  100 

53  =  125         10*  =  loog, 

O,  I,  2,  3  are  the  logarithms  of  the  respective  numbers  to  which  they  stand  opposite  in  the  several  equa- 
tions. 5  is  the  base  in  the  one  set,  and  ten  in  the  other.  In  any  one  of  the  above  quotations,  the  value 
of  the  second  member  depends  on  the  numerical  value  of  the  base  and  the  exponent  attached.  In  a  sys- 
tem of  logarithms,  any  number  above  I  may  be  taken  as  a  base,  and,  by  suitably  varying  the  exponent, 
the  base  being  unaltered,  all  possible  numbers  may  be  represented.  For  example,  lo^ -82000  represents 
the  number  6607,  and  3.82000  is  the  log.  of  this  number.  io3'7'*225  represents  the  number  5524,  and 
3.74225  is  the  log.  of  this  niimber.  It  means  that  10  must  be  raised  to  the  power  denoted  by  3.74225 
to  produce  5524.  In  all  practical  mathematics,  10  is  the  base.  The  system  is  called  the  Common,  or 
Brigg's  system,  and,  in  it,  all  numbers,  integral  or  fractional,  are  regarded  as  some  power  of  10.  10°  is 
no  power  of  10,  and  is  equal  to  10  divided  by  10,  or  to  I.  That  is,  the  log.  of  i  is  o.  All  numbers  be- 
tween I  and  10  have,  for  their  logarithm,  a  decimal  fraction  ;  all  numbers  between  10  and  100  have,  for 
their  logarithm,  i  -[-a  decimal ;  and  all  numbers  between  100  and  1000  have,  for  their  logarithm,  2  -(-  a 
decimal ;  and  so  on.  See,  page  I  of  the  table  of  logarithms,  in  column  headed  N,  that  numbers  be- 
tween I  and  10,  10  and  loo,  and  100  and  1000,  respectively,  fulfill  the  above  conditions.  The  log.  of  7, 
for  example,  is  .84510,  and  that  of  25  is  1-39794,  and  these  logs,  are  simply  exponents.  io''-845io  __  7^ 
and  ioi-^9794  =25,  signifying  that  10  must  be  raised  to  these  powers,  respectively,  to  produce  7  and  25. 

To  find  the  logarithms  of  numbers  over  100,  and  under  looo. — Look  opposite  the  number,  in  column 
headed  O,  and  find  the  logarithm.     The  log.  of  398,  page  7  of  the  table,  is  2.59988. 

.-To  find  the  logarithms  of  numbers  of  four  figures. — Look  under  caption  iV'for  the  first  three  'fig- 
ures of  the  number,  and  at  the  top  of  the  page  for  the  fourth  figure  ;  and  opposite  the  one  part  of  the 
number  and  under  the  other,  find  the  logarithm.  Thus,  the  log.  of  6982  is  3.84398.  The  decimal  part 
of  any  logarithm  is  called  the  mantissa,  and  the  integral  part,  the  characteristic.  In  the  log.  of  1840, 
which  is  3.26482,  3  is  the  characteristic,  and  .26482  is  the  mantissa.  The  characteristic  of  all  numbers 
between  I  and  10  is  o. 

To  find  the  logarithms  of  numbers  of  more  than  four  figures. — Find,  for  example,  the  logarithm  of 
248963.  On  page  4  we  find,  as  previously  directed,  the  «/rt«/wa*corresponding  to  the  first  four  figures, 
2489,  to  be  .39602  ;  and,  to  this  partial  mantissa,  there  must  be  an  addition  for  the  remaining  part  of 
the  number,  63.  And  since  this  addition  affects  only  the  decimal  part,  or  mantissa,  and  not  the  char- 
acteristic, 63,  the  remaining  part  of  the  number  must  be  regarded  as  a  decimal.  This  decimal,  .63,  we 
multiply  by  the  tabular  difference,  opposite  the  mantissa,  in  column  D,  which  is  17+,  or  17.5,  and  get 
.63  X  17.5  =  11.025,  giving  II  to  be  added  to  the  final  figures  of  the  partial  mantissa,  .39602,  already 
taken  out,  making  .39613  ;  and  the  characteristic  is  5,  being  always  one  less  than  the  number  of  figures 
in  the  integral  part  of  the  number  whose  logarithm  is  sought.     Thus,  the  log.  248963  =  5.39613. 

Required  the  logarithm  of  142967542.  The  mantissa  of  the  first  four  figures,  1429,  page  2  of  the 
table,  is  .15503,  and  the  tabular  difference  is  30+,  or  30.5.  This  multiplied  into  .67542,  the  remainder 
of  the  number  treated  as  a  decimal,  gives  20.6,  or  21,  to  be  added  to  the  terminal  figures  of  the  partial 
mantissa  already  taken  out,  making  .15524,  and  the  characteristic  is  8.  Thus,  the  log.  142967542  = 
8.15524.  In  making  additions  to  the  mantissa,  more  than  5  decimal  units  should  be  reckoned  i ,  less 
than  5  should  be  disregarded. 

For  the  same  figures,  in  the  same  order,  the  mantissa  is  the  same,  whatever  the  local  value  or  the  fig- 
ures. Thus,  the  mantissa  of  the  logs,  of  these  numbers,  viz.:  8328,  832.8,  83.28,  8.328,  .8328,  .08328. 
,008328,  etc.,  is  .92054,  the  same  for  each  of  the  numbers.  The  characteristic  of  the  first  is  3 ;  of  the 
second,  2  ;  of  the  third,  i ;  of  the  fourth,  o;  of  the  fifth,  — i ;  of  the  sixth,  —2 ;  of  the  seventh,  — 3.    The 


LOGARITHMS. 


23 


characteristic  of  a  decimal  is  always  negative,  and  numerically  one  more  than  the  number_of  noughts 
prefixed  to  the  decimal.  The  negative  sign  is  usually  vsrritten  over  the  characteristic,  thus  2, 6  9  8  9  7,  in 
the  log.  of  .05. 

Find  the  logarithm  of  .6423.  On  page  12  we  find  the  mantissa  to  be  .80774,  and  the  characteristic 
is  T,  making  T.  8  0  7  7  4,  for  the  log.  of  .6423. 

To  find  the  number  corresponding  to  a  given  logarithm. — What  is  the  number  whose  log.  Is 
1. 681^4?  Looking  on  page  I,  we  find  this  log.  opposite  48.  Hence,  48  is  the  number  whose  log.  is 
I. 68 I 24. 

Find  the  number  having  for  its  logarithm  2 , 3  6  3  0  5_.  Looking  on  page  4,  we  find  opposite  230 and 
under  7,  the  number  230.7,  the  answer  required^. 

Find  the  number  having  for  its  logarithm  2,64367.  Looking  for  the  nearest  mantissa  to  the  given 
one,  we  find  it,  page  8,  opposite  440  and  under  2,  to  be  .64365.  This  mantissa  we  subtract  from  the 
given  one,  and  divide  the  difference,  2,  by  the  tabular  diff^ence,  10,  and  get  .2.  Appending  this  to 
the  4402,  already  taken  out,  we  get  44022.  And  now,  as  the  characteristic  is  — 2,  we  prefix  one  o  to 
the  last  result,  and  get  .044022  for  the  required  number. 

What  is  the  number  having  for  its  logarithm  .29824?  The  nearest  mantissa  is  .29820,  page  3,  op- 
posite 198  and  under  7  ;  ,29824—  ,29820  =  4;  4-4-22,  the  tab.  difif.,  gives  1.8,  or  2  nearly.  Appending 
2  to  1987,  we  have  19872 — .  And,  since  there  is  no  characteristic,  the  integral  part  is  I.  Hence,  .29824 
= log.  I. 19872. 

;  Find  the  cube  root  of  .4986.  The  log.  of  .4986  (p.  9)  is  T.6  9  775,  This  we  divide  by  3,  the  in- 
dex of  the  required  root.  But  since  the  characteristic  is  negative,  while  the  mantissa  is  always  positive, 
we  cannot  directly  divide  the  logarithm  by  the  index  3.  But  T,6  9  7  7  S  ==  ^-f  2^.69775,  in  which  the 
characteristic  is  exactly  divisible  by  the  index  3.  Dividing,  we  get  T,?99  2  5,  We  now  find  the  num- 
ber corresponding  to  this  logarithm.  The  nearest  mantissa  is  .89922.'  Subtracting  it  from  the  given 
.89925,  we  get  3,  which,  divided  by  the  corresponding  tabular  diffe^nce,  §-f '  gives  5.  The  number 
corresponding  to  the  mantissa  .89925  is  79295.  To  this  prefix^'wic.  o,  to  correspond  to  the  negative 
characteristic,  and  the  cube  root  of  .4986  is  .-&7929§^    ^^  ,^  y-  ^7   <^J^ 

When  the  characteristic  is  negative,  and  not  divisible  by  me4nclex^of  any  root,  add  to  it  the  smallest  nega- 
tive number  that  will  render  it  divisible,  and  then  prefix  the  same  number,  with  a  plus  sign,  to  the  mantissa. 

What  is  the  5th  root  of  512.8?  The  log.  of  512.8  =  2.70995  ;  2,70995  -f-  5  "=  -54199  ;  the  nearest 
mantissa,  opposite  348  and  3,  is. 54195;  .54199 — .54195=4;  and  4 -r  12^-,  the  tab,  diff,,  gives  3  to 
to  be  apended  to  3.483,  making  3.4833,  for  the  5th  root  of  512.8. 


k 


5,if2^«02. 




T 

THE  COMMOH  OR  BRIGGS  LOGARITHMS 

—OF  THE—     . 

iT-A.rrxTie-A.ij  3sr"U-3iv^BEies 

FROM  1  TO  10000. 

I— 100. 

H 

loS 

If 

log 

ir 

log 

N 

log 

N 

log 

I 

0.  00  000 

21           I 

32   222 

41 

I.  61  278 

61        I 

78533 

81 

I.  90  849 

2 

0.  30  103 

22           I 

34  242 

42 

62325 

62        I 

79  239 

82 

I.  91  381 

3 

0.  47  712 

23          I 

36  173 

43 

63347 

63        I 

79  934 

83 

I.  91  908 

4 

0.  60  206  ■ 

24          I 

38021 

44 

64345 

64        I 

80618 

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208x7  20844  20871  20898  20925 

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21  085  2X  XX2  21  139  21  X65  21  I92 

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21  6x7  21  643  21  669  21  696  21  722 

26 

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21  880  21  906  21  932  21  958  2X  985 

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22  141  22  167  22  194  22  220  22  246 

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22  660  22  686  22  712  22  737  22  763 

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23  172  23  198  23  223  23  249  23  274 

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24920  24944  24969  24993  25018 

24+ 

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25  042  25  066  25  09X  25  115  25  139 

25  164  25  188  25  2X2  25  237  25  261 

24 

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25406  25431  25455  25479  25503 

24 

180 

25527  25551  25575  25600  25624 

25  648  25  672  25  696  25  720  25  744 

24 

181 

25  768  25  792  25  8x6  25  840  25  864 

25888  25912  25935  25959  25983 

24 

182 

26007  26031  26055  26079  26x02 

26  126  26  150  26  174  26  198  26  221 

24 

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26364  26387  26411  26435  26458 

24 

184 

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26  600  26  623  26  647  26  670  26  694 

23+ 

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23 

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27  068  27  091  27  114  27  138  27  x6i 

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27300  27323  27346  27370  27393 

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23 

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23 

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28  103  28  126  28  149  28  171  28  194 

282x7  28240  28262  28285  28307 

23 

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28330  28353  28375  28398  28421 

28443  28466  28488  28  511  28533 

23 

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28556  28578  28601  28623  28646 

286-68  28691  28713  28735  28758 

22+ 

194 

28  780  28  803  28  825  28  847  28  870 

28892  28914  28937  28959  28981 

22 

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29  003  29  026  29  048  29  070  29  092 

29  115  29137  29x59  29181  29203 

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22 

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22 

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29  667  29  688  29  7x0  29  732  29  754 

29776  29798  29  S20  29842  29863 

22 

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29885  29907  29929  29951  29973 

29994  300x6  30038  30060  30081 

22 

200 

30  103  30  12S  30  146  30  168  30  190 

30  211  30233  30255  30276  30298 

22 

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30  211  30233  30255  30276  30298 

22 

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30428  30449  30471  30492  30514 

22 

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30643  30664  30685  30707  30728 

21+ 

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30750  30771  30792  30814  30835 

30856  30878  30899  30920  30942 

21 

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30  963  30  984  31  006  31  027  31  048 

31  069  31  091  31  112  31  133  31  154 

21 

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31  175  31  197  31  218  31  239  31 260 

31  281  31  302  31  323  31  345  31  3^^ 

21 

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31  387  31  408  31  429  31  450  31  471 

31492  31  513  31534  31555  31576 

21 

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31  597  31  618  31  639  31  660  31  681 

31  702  31  723  31  744  31  765  31  785 

21 

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31  806  31  827  31  848  31  869  31  890 

31  911  31931  31952  31973  31994 

21 

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32015  32035  32056  32077  32098 

32  1X8  32  139  32  160  32  181  32  20I 

21 

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32  222  32  243  32  263  32  284  32  305 

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21 

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32  428  32  449  32  469  32  490  32  510 

32531  32552  32572  32593  32613 

20+ 

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32  634  32  654  32  675  32  695  32  715 

32736  32  756  32777  32797  32818 

20 

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32  838  32  858  32  879  32  899  32  919 

32  940  32  960  32  980  33  001  33  021 

20 

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33  041  33  062  33  082  33  102  33  122 

33  143  33  ^^3  33  ^^3  33  203  33  224 

20 

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33  244  33  264  33  284  33  304  33  325 

33  345  33  365  33  385  33  405  33  425 

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216 

33  445  33  465  33  486  33  506  33  526 

33  546  33  566  33  586  33  606  33  626 

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33  646  33  666  33  686  33  706  33  726 

33  746  33  766  33  786  33  806  33  826 

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33  945  33  9^5  33  985  34  oo5  34  025 

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20 

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20 

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34  439  34  459  34  479  34  498  34  5^8 

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34635  34655  34674  34694  34713 

34  733  34  753  34  772  34  792  34  811 

19+ 

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34928  34  947  34967  34986  35005 

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35  122  35  141  35  160  35  180  35  199 

19 

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35  315  35  334  35  353  35  372  35  392 

19 

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35  411  35  430  35  449  35  468  35  488 

35  507  35  526  35  545  35  564  35  583 

19 

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35  603  35  622  35  641  35  660  35  679 

35  698  35  717  35  736  35  755  35  774 

19 

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35  793  35813  35832  35851  35870 

35  889  35  908  35  927  35  946  35  965 

19 

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35984  36003  36021  36040  36059 

36078  36097  36  116  36135  36154 

19 

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36173  36192  36  211  36229  36248 

36267  36286  3630^  36324  36342 

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36361  36380   36399  36418   36436 

36455  36474  36493  36  511  36530 

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36549  36568  36,586  36605  36624 

36642  36661  36680  36  6g8   36717 

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36  T36  36  754  36  773  36  791  36  810 

36829  36847  36866  36884  36903 

19 

234 

36922  36940  36959  36977  36996 

37014  37033  37051  37070  37088 

18+ 

235 

37  107  37  125  37  144  37  162  37  181 

37  199  37  218  37  236  37  254  37  273 

18 

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37291  37310  37328  37346  37365 

37383  37401  37420  37438  37  457 

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237 

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37566  37585  37603  37621  37639 

18 

238 

37  658  37  676  37  694  37  712  37  731 

37749,37767  37785  37803  37822 

18 

239 

37  840  37  858  37  876  37  894  37  912 

37  931  37  949  37  967  37  985  38  003 

18 

240 

38021  38039  38057  38075  38093 

38  112  38  130  38  148  38  166  38  184 

18 

241 

38  202  38  220  38  238  38  256  38  274 

38292  38310  38328  38346   38364 

18 

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38382  38399  38417  38435  38453 

38471  38489  38507  38525  38543 

18 

243 

38561  38578  38596  38614  38632 

38  650  38  668  38  686  38  703  38  721 

18 

244 

38739  38757  38775  38792  38810 

38828  38846  38863  38881  38899 

18 

245 

38917  38934  38952  38970  38987 

39005  39023  39041  39058  39076 

18 

246 

39094  39  III  39  129  39  146  39  164 

39182  39199  39217  39235  39252 

18 

247 

39270  39287  39305  39322  39340 

39358  39  375  39  393  39  4io  39428 

18 

248 

39445  39463  39480  39498  39515 

39  533  39550  39568  39585  39602 

17+ 

249 

39620  39637  39655  39672  39690 

39  707  39  724  39  742  39  759  39  777 

17 

250 

39  794  39  811  39829  39846  39863 

39.881  39898  39915  39  933   39950 

17 

N 

0    12    3    4 

5    6    7    8    9 

250 

251 
252 

253 
254 

255 
256 

257 
258 

259 

260 

26l 

262 

263 
264 

265 

266 

267 

268 

269 

270 

271 
272 
273 
274 

275 
276 

277 
278 
279 

280 
281 
282 
283 
284 

285 
286 
287 
288 
289 

290 

291 
292 

293 
294 

295 
296 

297 
298 

299 

300 


39  794 
39967 

40  140 
40312 
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40  824 

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41  162 

41330 

41  497 
41  664 
41  830 

41  996 

42  160 

42325 
42488 

42  651 
42813 
42975 

43  136 
43  297 
43  457 
43616 

43  775 

43  933 
44091 
44248 
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44560 

44716 
44871 
45025 
45  179 
45332 

45  484 
45  637 

45788 

45  939 

46  090 

46  240 
46389 

46538 
46687 

46  835 

46  982 

47  129 
47  276 
47422 
47567 
47  712 

0 


398II 
39985 
40157 
40329 

40  500 

40  671 

40  841 

41  010 

41 179 
41347 

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41  681 

41  847 

42  012 
42  177 

42341 
42504 
42  667 

42  830 
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43  152 
43313 
43  473 
43  632 

43  791 

43949 

44  107 

44  264 
44420 
44576 

44731 
44886 

45  040 
45  194 
45  347 
45  500 
45652 
45803 

45  954 

46  io5 

46255 
46  404 

46553 

46  702 
46850 

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47  144 
47  290 
47436 
47582 

47727 


39  829  39  846  39  863 

40  002  40  019  40  037 
40  175  40  192  40  209 
40346  40364  40381 
40518  40535  40552 
40  688  40  705  40  722 

40  858  40  875  40  892 

41  027  41  044  41  061 
41  196  41  212  41  229 
41  Z^2,   41  380  41  397 

41  531  41  547  41  564 
41  697  41  714  41  731 

41  863  41  880  41  896 

42  029  42  045  42  062 
42  193  42  210  42  226 

42357  42374  42390 
42521  42  537  42  553 
42  684  42*700  42  716 

42  846  42  862  42  878 

43  008  43  024  43  040 

43  169  43  185  43  201 

43329  43345  43361 

43  489  43505  43  521 

43  648  43  664  43  680 

43  807  43  823  43  838 

43  96S  43  981  43  996 
44122  44138  44154 
44279  44295  44  311 
44436  44451  44467 
44592  44607  44623 

44  747  44762  44778 
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45  056  45071  45  086 
45  209  45  225  45  240 
45  362  45  378  45  393 

45515  45530  45  545 
45  667  45  682  45  697 
45  818  45  834  45  849 

45  969  45  984  46  000 

46  120  46  135  46  i5o 

46  270  46  285  46  300 
46  419  46  434  46  449 
46  568  46  583  46  598 
46  716  46  731  46  746 

46  864  46  879  46  894 

47  012  47  026  47  041 
47  159  47  173  47  188 
47305  47319  47  334 
47451  47465  47480 
47596  47  611  47  625 

47  741  47  756  47  770 


39881  39898  39915 
40  054  40  071  40  088 
40  226  40  243  40  261 
40398  40415  40432 
40569  40586  40603 

40  739  40  756  40  773 

40  909  40  926  40  943 

41  078  41  095  41  III 
41  246  41  263  41  280 
41  414  41  430  41  447 

41  581  41  597  41  614 
41  747  41  764  41  780 

41  913  41  929  41  946 

42  078  42  095  42  III 
42  243  42  259  42  275 

42  406  42  423  42  439 

42  570  42  586  42  602 

42  732  42  749  42  765 

42  894  42  911  42  927 

43  056  43  072  43  088 

43217  43233  43  249 
43  377  43  393  43  409 
43  537  43  553  43  569 
43696  43  712  43  727 

43  854  43  870  43  886 

44  012  44  028  44  044 

44  170  44  185  44201 
44326  44342  44358 
44483  44498  44514 
44638  44654  44669 

44793  44809  44824 

44948  44963  44  979 

45  102  45  117  45  ^Z?, 
45  255  45  271  45  286 
45  408  45  423  45  439 

45561  45576  45591 

45  712  45  728  45  743 

45  864  45  879  45  894 

46  oi5  46  030  46  045 
46  165  46  180  46  195 

46315  46330  46345 

46  464  46  479  46  494 

46  613  46  627  46  642 

46  761  46  776  46  790 

46  909  46  923  46  938 

47  056  47  070  47  085 
47  202  47  217  47  232 
47  349  47  363  47  378 
47  494  47  509  47  524 
47  640  47  654  47  669 

47  784  47  799  47  813 


39  933 

40  106 
40  278 

40449 
40  620 

40790 

40  960 

41  128 
41  296 
41  464 

41  631 
41  797 

41  963 

42  127 
42  292 

42455 
42619 

42  781 
42943 

43  104 

43  265 
43425 
43584 
43  743 

43  902 

44059 
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44  373 
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44  994 

45  148 
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45  454 
45  606 
45  758 

45  909 

46  060 
46  210 

46359 
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47  100 
47  246 
47392 
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39950 
40  123 
40  295 
40  466 
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40  807 
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41  145 
41  313 
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41  647 
41  814 

41  979 

42  144 
42  308 

42  472 
42  635 

42  797 
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43  120 

43  281 
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43  600 
43  759 
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44075 
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45  010 
45  163 
45  317 
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45  621 
45  773 

45  924 

46  075 
46  225 

46374 
46532 
46  672 

46  820 
46967 

47  114 
47  261 

47  407 
47  553 
47698 

47842 
9 


N 

0    12    3    4 

5    6    7    8    9 

D 

300 

47  712  47  727  47  741  47  75^  47  77© 

47  784  47  799  47  813  47  828  47  842 

14+ 

301 

47857  47871  47885  47900  47914 

47  929  47  943  47  958  47  972  47  986 

14 

302 

48  001  48  015  48  029  48  044.  48  058 

48  073  48  087  48  loi  48  116  48  130 

14 

303 

48  144  48  159  48  173  48  187  48  202 

48  216  48  230  48  244  48  259  48  273 

14 

304 

48287  48302  48316  48330  48344 

48359  48373  48387  48401  48416 

14 

305 

48  430  48  444  48  458  48  473  48  487 

48501  48515  48530  48544  48558 

14 

306 

48572  48586  48601  48615  48629 

48643  48657  48671  48686  48700 

14 

307 

48  714  48  728  48  742  48  756  48  770 

48785  48799  48813  48827  48841 

14 

308 

48855  48869  4888s   48897  48  911 

48  926  48  940  48  954  48  968  48  982 

14 

309 

48996  49010  49024  49038  49052 

49  066  49  080  49  094  49  108  49  122 

14 

310 

49  136  49  150  49  164  49  ^78  49  192 

49  206  49  220  49  234  49  248  49  262 

14 

311 

49276  49290  49304  49318  49332 

49346  49360  49  374  49388  49402 

14 

312 

49415  49429  49  443  49457  49471 

49485  49  499  49513  49527  49541 

14 

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49  554  49568  49582  49596  49610 

49624  49638  49651  49665  49679 

14 

314 

49  693  49  707  49  721  49  734  49  748 

49  762  49  776.  49  790  49  803  49  817 

14 

315 

49831  49845  49859  49872  49886 

49900  49  9U  49927  49941  49955 

14 

3^6 

49969  49982  49996  50010  50024 

50037  50051  50065  50079  50092 

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317 

50  106  50  120  50  133  50  147  50  161 

50174  50188  50202  50215  50229 

14 

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50243  50256  50270  50284  50297 

50  311  50325  50338  50352  50365 

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319 

50379  50393  50406  50420  50433 

50447  50461  50474  50488  50501 

14 

320 

50515  50529  50542  50556  50569 

50583  50596  50610  50623  50637 

14 

321 

50651  50664  50678  50691  50705 

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13+ 

322 

50786  50799  50813  50826  50840 

50853  50866  50880  50893  50907 

13+ 

323 

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50987  51  001  51  014  51028  51  041 

13 

324 

51  055  51  068  51  081  51  095  51  108 

51  121  51  135  51  148  51 162  51  175 

13 

325 

51  188  51  202  51  215  51  228  51  242 

51  255  51  268  51  282  51  295  51  308 

13 

326 

51322  51335  51348  51362  51375 

51388  51402  51  415  51428  51  441 

13 

327 

51455  51468  51  481  51495  51508 

51  521  51534  51548  51  561  51574 

13 

328 

51  587  51  601  51  614  51  627  51  640 

51  654  51  667  51  680  51  693  51  706 

13 

329 

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51  786  51  799  51  812  51  825  51  838 

13 

330 

51  851  51865  51878  51  891  51904 

51917  51930  5x943  51957  51970 

13 

331 

51983  51996  52009  52022  52035 

52  048  52  061  52  075  52  088  52  lOI 

13 

332 

52  114  52  127  52  140  52  153  52  166 

52  179  52  192  52  205  52  218  52  231 

13 

333 

52  244  52  257  52  270  52  284  52  297 

52310  52323  52336  52349  52362 

13 

334 

52  375  52  388  52  401  52  414  52  427 

52440  52453  52466  52479  52492 

13 

335 

52504  52517  52530  52543  52556 

52569  52582  52595  52608  52621 

13 

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52  634  52  647  52  660  52  673  52  686 

52  699  52  711  52  724  52  737  52  750 

13 

337 

52  763  52  776  52  789  52  802  52  815 

52827  52840  52853  52866  52879 

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338 

52892  52905  52917  52930  52943 

52956  52969  52982  52994  53007, 

13 

339 

53020  53033  53046  53058  53071 

53  084  53  097  53  "o  53  122  53  135 

13 

340 

53  148  53  161  53  173  53  186  53  199 

53  212  53  224  53  237  53  250  53  263 

13 

341 

53275  53288  53301  53314  53326 

53  339  53352  53364  53377  53390 

13 

342 

53403  53  4^5  53428  53441  53453 

53466  53479  53491  53504  53517 

13 

343 

53529  53542  53555  53567  53580 

53  593  53605  53618  53631  53643 

13 

344 

53  656  53  668  53  681  53  694  53  706 

53  719  53  732  53  744  53  757  53  769 

13 

345 

53  782  53  794  53  807  53  820  53  832 

53845  53857  53870  53882  53895 

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53908  53920  53  933  53  945  53  958 

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347 

54033  54045  54058  54070  54083 

54095  54108  54120  54133  54145 

12+ 

348 

54158  54170  54183  54195  54208 

54220  54233  54245  54258  54270 

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349 

54283  54295  54307  54320  54332 

54  345  54  357  54  37©  54382  54  394 

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350 

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351 

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54  593  54605  54617  54630  54642 

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352 

54654  54667  54679  54691  54704 

54716  54728  54741  54  753  54765 

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353 

54777  54790  54802  54814  54827 

54839  54851  54864  54.876  54888 

12 

354 

54900  54913  54925  54  937  54  949 

54962  54  974  54986  54998  55  on 

12 

355 

55  023  55  035  55  047  55  060  55  072 

55084  55096  55  108  55  1.21  55  133 

12 

356 

55  145  55  157  55  ^^9   55  182  55  194 

55206  55218  55230  55242  55  255. 

12 

357 

55267  55279  55291  55303  55315 

55328  55340  55352  55364  55376 

12 

358 

55  Z^^  SS  400  55  413  55  425  55  437  ' 

55  449  55  461  55  473  55  485  55  497 

12 

359 

55509  55522  55534  55546  55558 

55570  55582  55594  55606  55618 

12 

360 

55  630  55  642  55  654  55  666  55  678 

55  691  55  703  55  715  55  727  55  739 

12 

361 

55751  55763  55  775  55  787  55  799 

55  811  55823  55835  55847  55859 

12 

362 

55871  55883  55895  55907  55919 

55931  55  943  55955  55967  55  979 

12 

363 

55991  56003  56015  56027  56038 

56050  56062  56074  56086  56098 

12 

364 

56  no  56  122  56  134  56  146  56  158 

56  170  56  182  56  194  5*6  205  56  217 

12 

365 

56  229  56  241  56  253  56  265  56  277 

56289  56  301  56  312  56324  56336 

12 

366 

56348  56360  56372  56384  56396 

56407  56419  56431  56443  56455 

12 

367 

56467  56478  56490  56502  56514 

56526  56538  56549  56561  56573 

12 

368 

56585  56597  56608  56620  56632 

56644  56656  56667  56679  56691 

12 

369 

56703  56714  56726  56738  56750 

56761  56773  56785  56797  56808 

12 

370 

56820  56832  56844  56855  56867 

56879  56891  56902  56914  56926 

12 

371 

56937  56949  56961  56972  56984 

56996  57008  57019  57031  57043 

12 

372 

57  054  57  066  57  078  57  089  57  loi 

57  113  57  124  57  136  57  148  57  159 

12 

373 

57  171  57  ^^2,   57  i94  57  206  57  217 

57  229  57  241  57  252  57  264  57  276 

12 

374 

57287  57299  57310  57322  57334 

57  345  57  357  57  368  57380  57  392 

12 

375 

57  403  57  415  57  426  57  438  57  449 

57461  57  473  57484  57496  57507 

12 

376 

57519  57530  57542  57  553  57565 

57576  57588  57600  57  611  57623 

11+ 

377 

57  634  57  646  57  657  57  669  57  680 

57  692  57  703  57  715  57  726  57  738 

"+ 

378 

57  749  57  761  57  772  57  784  57  795 

57807  57818  57830  57841  57852 

"+ 

379 

57864  57875  57887  57898  57910 

57921  57  933  57  944  57  955  57  967 

380 

57978  57990  58001  58013  58024 

58035  58047  58058  58070  58081 

381 

58092  58  104  58  115  58  127  58  138 

58  149  58  161  58  172  58  184  58  195 

382 

58  206  58  218  58  229  58  240  58  252 

58263  58274  58286  58297  58309 

383 

583^0  58331  58343  58354  58365 

58377  58388  58399  58410  58422 

384 

58433  58444  58456  58467  58478 

58490  58501  58512  58524  58535 

385 

58546  58557  58569  58580  58591 

58602  58614  58625  58636  58647 

386 

58659  58670  58681  58692  58704 

58715  58726  58737  58749  58760 

387 

58771  58782  58794  58805  58816 

58827  58838  58850  58861  58872 

388 

58883  58894  58906  58917  58928 

58939  58950  58961  58973  58984 

389 

58995  59006  59017  59028  59040 

59051  59062  59073  59084  59095 

390 

59  106  59  118  59  129  59  140  59  151 

59162  59173  59184  59195  59207 

391 

59218  59229  59240  59251  59262 

59273  59284  59295  59306  59318 

392 

59329  59340  59351  59362  59373 

59384  59  395  59406  59417  59428 

393 

59439  59450  59461  59472  59483 

59494  59506  59517  59528  59  539 

394 

59550  59561  59572  59583  59594 

59605  59616  59627  59638  59649 

395 

59660  59671  59682  59693  59704 

59  715  59  726  59  737  59  748  59  759 

396 

59770  59780  59791  59802  59813 

59824  59835  59846  59857  59868 

397 

59879  59890  59901  59912  59923 

59  934  59945  59956  59966  59  977 

398 

59988  59999  60010  60021  60032 

60  043  60  054  60  065  60  076  60  086 

399 

60  097  60  108  60  119  60  130  60  141 

60  152  60  163  60  173  60  184  60  195 

400 

60206  60217  60228  60239  60249 

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N 

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5    6     7    8    9 

N 

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400 

60  206  60  217  60  228  60  239  60  249 

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401 

60314  60325  60:^^6  60347  60358 

60369  60379  60390  60401 

60  412 

II 

402 

60423  60433  60444  60455  60466 

60477  60487  60498  60509 

60  520 

II 

403 

60531  60541  60552  60563  60574 

60584  60595  60606  60617 

60  627 

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404 

606^8  60649  60660  60670  60681 

60692  60703  60713  60724 

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60  746  60  756  60  767  60  778  60  788 

60799  60810  60821  60831 

60  842 

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406 

60853  6086^   60874  60885  60895 

60906  60917  60927  60938 

60  949 

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407 

60959  60970  60981  609.91  61002 

61  013  61  023  61  034  61  045 

61  055 

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408 

61  066  61  077  61  087  61  098  61  109 

61  119  61  130  61  140  61  151 

61  162 

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409 

61  172  61  183  61  194  61  204  61  2l5 

61  225  61  236  61  247  61  257 

61 268 

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410 

61  278  61  289  61  300  61  310  61  321 

61  331  61342  61  352  61  s6s 

61  374 

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411 

61  384  61  395  61  405  61  416  61  426 

61  437  61  448  61  458  6x  469 

61  479 

II 

412 

61  490  61  500  61  511  61  521  61  532 

61  542  61  553  61  563  61  574 

61584 

10+ 

413 

61  595  61  606  61  616  61  627  61  637 

61  648  61  658  61  669  61  679 

61  690 

10+ 

414 

61  700  61  711  61  721  61  731  6i  742 

61  752  61  763  61  773  61  784 

61  794 

10+ 

415 

61  805  61  815  61  826  61  836  61  847 

61857  61868  61878  61888 

61  899 

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416 

61  909  61  920  61  930  61  941  61  951 

61  962  61  972  61  982  61  993 

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417 

62  014  62  024  62  034  62  045  62  055 

62  066  62  076  62  086  62  097 

62  107 

10 

418 

62  118  62  128  62  138  62  149  62  159 

62  170  62  180  62  190  62  201 

62  211 

10 

419 

62  221  62  232,  62  242  62  252  62  263 

62  273  62  284  62  294  62  304 

62315 

10 

420 

62325  62335  62346  62356  62^66 

62  377  62  387  62  397  62  408 

62  418 

10 

421 

62  428  62  439  62  449  62  459  62  469 

62  480  62  490  62  500  62  511 

62  521 

10 

422 

62531  62542  62552  62562  62572 

62  583  62  593  62  603  62  613 

62  624 

10 

423 

62  634  62  644  62  655  62  665  62  675 

62  685  62  696  62  706  62  716 

62  726 

10 

424 

62  737  62  747  62  757  62  767  62  778 

62  788  62  798  62  808  62  8i8 

62829 

10 

425 

62  839  62  849  62  859  62  870  62  880 

62  890  62  900  62  910  62  921 

62931 

10 

426 

62941  62951  62961  62972  62982 

62992  63002  63012  63022 

63033 

10 

427 

^3  043  ^3  053  (^3  °^3   63  073  6s  083 

63094  63  104  63  114  6s  124 

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428 

63  144  6^  155  63  i65  63  175  6s  185 

6s  195  63  205  6s  215  63  225 

63236 

10 

429 

63  246  6s  256  6s  266  6s  276  6s  286 

63296  6s  306   63317  63327 

63  337 

10 

430 

63347  63357  6s  367   63377  63387 

63397  63407  63417  63428  63438 

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68  886  68895  68904  68913  68922 

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69  064  69  073  69  082  69  090  69  099 

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69  152  69  161  69  170  69  179  69  188 

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69  197  69  205  69  214  69  223  69  232 

69  241  69  249  69  258  69  267  69  276 

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494 

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69417  69425  69434  69443  69452 

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9 

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498 

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69  767  69  715  69  784  69  793  69  801 

9 

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500 

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70027  70036  70044  70053  70062 

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70  114  70122  70  131  70140  70148 

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70286  70295  70303  70312  70321 

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505 

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70372  70381  70389  70398  70406 

9 

506 

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70458  70467  70475  70484  70492 

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507 

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70544  70552  70561  70569  70578 

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512 

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71  012  71020  71029  71037  71046 

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71  096  71  105  71  113  71  122  71  130 

71  139  71  147  71  155  71  164  71  172 

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515 

71  181  71  189  71  198  71 206  71  214 

71  223  71  231  71 240  71  248  71  257 

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71  265  71  273  71  282  71  290  71  299 

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71  391  71399  71408  71  416  71425 

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519 

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71  559  71  567  71  575  71  584  71  592 

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520 

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71  725  71  734  71  742  71  750  71  759 

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525 

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72  222  72  230  72  239  72  247  72  255 

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535 

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536 

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73  440  73  448  73  456  73  464  73  472 

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73  600  73  608  73  616  73  624  73  632 

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605 
606 
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78  176  78  183  78  190  78  197  78  204 
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78  211  78219  78226  78233  78240 
78283  78290  78297  78305  78312 
78355  78362  78369  78376  78383 
78426  78433  78440  78447  78455 
78497  78504  78512  78519  78526 

610 

611 
612 

613 
614 

78533  78540  78547  78554  78561 
78604  78  611  78618  78625  'jsess 
78675  78682  78689  78696  78704 
78746  78753  78760  78767  78774 
78817  78824  78831  78838  78845 

78569  78576  78583  78590  78597 
78640  78647  78654  78661  78668 
78  711  78718  78725  78732  78739 
78781  78789  78796  78803  78810 
78852  78859  78866  78873  78880 

615 
616 
617 
618 
619 

78  888  78  895  78  902 
78958  78965  78972 
79029  79036  79043 
79099  79  106  79  113 

79  169  79  176  79  ^^3 

78  909  78  916 
78979  78986 
79050  79057 

79  120  79  127 
79  190  79  197 

78923  78930  78937  78944  78951 
78993  79000  79007  79014  79021 
79064  79071  79078  79085  79092 
79  134  79  141  79  148  79  ^55  79  162 
79204  79  211  79218  79225  79232 

620 

621 
622 
623 
624 

79  239  79  246  79  253 
79309  79316  79323 
79  379  79386  79  393 
79  449  79456  79463 
79518  79525  79532 

79  260^79  267 

79330*79337 
79400  79407 

79470  79  477 
79  539  79546 

79274  79281  79288  79295  79302 
79  344  79351  79358  79365  79372 
79414  79421  79428  79435  79442 
79484  79491  79498  79505  79  511 
79  553  79560  79567  79  574  79  581 

625 
626 
627 
628 
629 

79588  79595  79602 
79657  79664  79671 
79727  79734  79741 
79  796  79803  79810 
79865  79872  79879 

79  609  79  616 
79678  79685 
79  748  79  754 
79817  79824 
79886  79893 

79623  79630  79637  79644  79650 
79692  79699  79706  79713  79720 
79  761  79  768  79  775  79  782  79  789 
79831  79837  79844  79851  79858 
79900  79906  79913  79920  79927 

630 

631 
632 

634 

79  934  79941  79948  79955  79962 
80003  80010  80017  80024  80030 
80072  80079  80085  80092  80099 

80  140  80  147  80  154  80  161  80  168 
80209  80216  80223  80229  80236 

79969  79  975  79982  79989  79996 
80037  80044  80051  80058  80  o65' 
80  106  80  113  80  120  80  127  80  134 
80  175  80  182  80  188  80  195  80  202 
80243  80250  80257  80264  80271 

635 
636 

637 
638 

639 

80  277  80  284  80  291 
80346  80353  80359 
80414  80  421  80  428 
80482  80489  80496 
80550  80557  80564 

80  298  80  305 
80366   80373 
80434  80441 
80502  80  509 
80570  80577 

80312  80318  80325  80332  80339 
80380  80387  80393  80400  80407 
80  448  80  455  80  462  80  468  80  475 
80516  80523  80530  80536  80543 
80584  80591  80598  80604  80  611 

640 

641 
642 

643 
644 

80618  80625  80632  80638  80645 
80686  80693  80699  80706  80713 
80  754  80  760  80  767  80  774  80  781 
80821  80828  80835  80841  80848 
80889  80895  80902  80909  80916 

80652  80659  80665  80672  80679 
80  720  80  726  80  733  80  740  80  747 
80787  80794  80801  80808  80814 
80855  80862  80868   80875  80882 
80922  80929  80936  80943  80949 

645 
646 

647 
648 
649 

80  956  80  963  80  969 

81  023  81  030  81  037 
81  090  81  097  81  104 
81  158  81  164  81  171 
81  224  81  231  81  238 

80  976  80  983 

81  043  81  050 
81  III  81  117 
81  178  81  184 
81  245  81  251 

80990  80996  81003  81  010  81  017 
81  057  81  064  81  070  81  077  81  084 
81  124  81  131  81  137  81  144  81  151 
81  191  81  198  81  204  81  211  81  218 
81  258  81  265  81  271  81  278  81  285 

650 

81  291  81  298  81  305 

81  311  81  318 

81325  81  331  81338  81345  81  351 

7 

N 

0    12 

3    4 

5    6    7    8    9 

N 

0    12    3    4 

5    6    7    8    9 

D 

650 

651 
652 

653 
654 

81  291  81 298  81  305  81  311  81  318 
81358  81365  81  371  81378  81385 
81  425  81  431  81  438  81  445  81  451 
81  491  81  498  81  505  81  511  81  518 
81558  81564  81  571  81578  81584 

81325  81  331  81338  81345  81  351 
81  391  81398  81405  81411  81418 
81  458  81  465  81  471  81  478  81  485 
81  525  81 531  81  538  81  544  81 551 
81  591  81  598  81  604  81  611  81  617 

655 
656 

657 
658 

659 

Bl  624  81  631  81  637  81  644  81  651 
81  690  81  697  81  704  81  710  81  717 
8x757  81763  81770  81  776  81783 
81  823  81  829  81  8^6  81  842  81  849 
81  889  81  895  81  902  81  908  81  915 

81  657  81  664  81  671  81  677  81  684 
81  723  81  730  81  737  81  743  81  750 
81 790  81  796  81  803  81  809  81  816 
81  856  81  862  81  869  81  875  81  882 
81  921  81  928  81  935  81  941  81  948 

660 

661 
662 

664 

81  954  81  961  81  968  81  974  81  981 

82  020  82  027  82  033  82  040  82  046 
82  086  82  092  82  099  82  105  82  112 
82  151  82  158  82  164  82  171  82  178 
82  217  82  223  82  230  82  236  82  243 

81  987  81  994  82  000  82  007  82  014 

82  053  82  060  82  066  82  073  82  079 
82  119  82  125  82  132  82  138  82  145 
82  184  82  191  82  197  82  204  82  210 
82  249  82  256  82  263  82  269  82  276 

6 

6+ 

665 
666 
667 
668 
669 

82  282  82  289  82  295  82  302  82  308 
82347  82354  82360  82367  82373 
82  413  82  419  82  426  82  432  82  439 
82  478  82  484  82  491  82  497  82  504 
82543  82549  82556  82562  82569 

82  315  82  321  82  328  82  334  82  341 
82  380  82  387  82  393  82  400  82  406 
82445  82452  82458  82465  82471 
82510  82517  82523  82530  82536 
82575  82582  82588  82595  82601 

6+ 
6+ 
6+ 
6+ 
6+ 

670 

671 

672 

673 
674 

82  607  82  614  82  620  82  627  82  6^3 
82  672  82  679  82  685  82  692  82  698 
82  737  82  743  82  750  82  756  82  763 
82802  82808  82814  82821  82827 
82866   82872  82879  82885  82892 

82  640  82  646  82  653  82  659  82  666 
82  705  82  711  82  718  82  724  82  730 
82  769  82  776  82  782  82  789  82  795 
82  834  82  840  82  847  82  853  82  860 
82898  82905  82  911  82918  82924 

6+ 

6+ 

6+ 

6 

6 

675 
676 

677 
678 
679 

82  930  82  937  82  943  82  950  82  956 

82  995  83  001  83  008  83  014  83  020 
8s  059  83  065  83  072  83  078  83  085 

83  123  83  129  83  136  83  142  83  149 
83  187  83  193  83  200  83  206   83.213 

82  963  82  969  82  975  82  982  82  988 

83  027  83  033  83  040  83  046  83  052 
83  091  83  097  83  104  83  no  83  117 
83  155  83  161  83  168  83  174  83  181 
83  219  83  225  83  232  83  238  83  245 

6 
6 
6 
6 
6 

680 

681 
682 
683 
684 

83  251  83  257  83  264  83  270  83  276 
83315  83321  83327  83334   83340 
8337^   83385  83391  83398  83404 
83  442  83  448  83  455  83  461  83  467 
83506  83512  83518  83525  83531 

83  283  83  289  83  296  83  302  83  308 

^3  347  83  353  ^3  359  ^3  3^^  ^3  372 
83  410  83  417  83  423  83  429  83  436 
^3  474  ^3  480  83  487  83  493  83  499 
83  537  ^3  544  ^3  55°  ^3  55^  S3  563 

6 
6 
6 
6 
6 

685 
686 
687 
688 
689 

^3  569  83  575  83  582  83  588  83  594 
83632  83639  83645  83651  83658 
83  696  83  702  83  708  83  715  83  721 
83759  83765  83771  83778  83784 
83  822  83  828  83  835  83  841  83  847 

83  601  83  607  83  613  83  620  83  626 
83  664  83  670  83  677  83  683  83  689 
S3  727  83  734  83  740  83  746  83  753 
83  790  83  797  83  803  83  809  83  816 
83  853  S3  860  83  866  83  872  83  879 

6 
6 
6 
6 
6 

690 

691 
692 

693 
694 

83  885  83  891  83  897  83  904  83  910 

83  948  83  954  83  960  83  967  83  973 

84  01 1  84017  84023  84029  84036 
84073  84080  84086  84092  84098 
84  136  84  142  84  148  84  155  84  i6i 

83  916  83  923  83  929  83  935  83  942 
83979  83985  83992  83998  84004 
84042  84048  84055  84061  84067 
84105  84  III  84  117  84123  84130 

84  167  84  173  84  180  84  186  84  192 

6 
6 
6 
6 
6 

695 
696 
697 
698 
699 

84198  84205  84  211  84217  84223 
84261  84267  84273  84280  84286 
84323  84330  84336  84342  84348 
84386  84392  84398  84404  84410 
84448  84454  84460  84466  84473 

84  230  84  236  84  242  84  248  84  255 
84292  84298  84305  84311  84317 
84354  84361  84367  84373  84379 
84417  84423  84429  84435  84442 
84479  84485  84491  84497  84504 

6 
6 
6 
6 
6 

700 

84510  84516  84522  84528  84535. 

84541  84547  84553  84559  84566 

6 

- 

0    12    3    4 

5    6     7    8     9 

N 

0    12    3    4 

5    6    7    8    9 

D 

700 

84510  84516  84522  84528  84535 

84541  84547  84553  84559  84566 

6 

701 

84572  84578  84584  84590  84597 

84603  84609  84615  84621  84628 

6 

702 

84634  84640  84646  84652  84658 

84665  84671  84677  84683  84689 

6 

703 

84696  84702  84708  84714  84720 

84726  84733  84739  84745  84751 

6 

704 

84757  84763  84770  84776  84782 

84788  84794  84800  84807  84813 

6 

705 

84819  84825  84831  84837  84844 

84850  84856  84862  84868  84874 

6 

706 

84880  84887  84893  84899  84905 

84  911  84917  84924  84930  84936 

6 

707 

84942  84948  84954  84960  84967 

84973  84979  84985  84991  84997 

6 

708 

85  003  85  009  85  016  85  022  85  028 

85  034  85  040  85  046  85  052  85  058 

6 

709 

85  065  85  071  85  077  85  083  85  089 

85  095  85  loi  85  107  85  114  85  120 

6 

710 

85  126  85  132  85  138  85  144  85  150 

85  156  85  163  85  169  85  175  85  181 

6 

711 

85  187  85  193  85  199  85  205  85  211 

85  217  85  224  85  230  85  236  85  242 

6 

712 

85  248  85  254  85  260  85  266  85  272 

85  278  85  285  85  291  85  297  85  303 

6 

713 

85309  85315  85321  85327  85333 

85  339  85  345  85  352  85  358  85  364 

6 

714 

85  370  85  376  85  382  85  388  85  394 

85  400  85  406  85  412  85  418  85  425 

6 

715 

85  431  85  437  85  443  85  449  85  455 

85  461  85  467  85  473  85  479  85  485 

6 

716 

85  491  85  497  85  503  85  509  85  516 

85  522  85  528  85  534  85  540  85  546 

6 

717 

85552  85558  85564  85570  85576 

85  582  85  588  85  594  85  600  85  606 

6 

718 

85  612  85  618  85  625  85  631  85  637 

85  643  85  649  85  655  85  661  85  667 

6 

719 

85  673  85  679  85  685  85  691  85  697 

85  703  85  709  85  715  85  721  85  727 

6 

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85  733  85  739  85  745  85  751  85  757 

85  763  85  769  85  775  85  781  85  788 

6 

721 

85  794  85  800  85  806  85  812  85  818 

85  824  85  830  85  836  85  842  85  848 

6 

722 

85  854  85  860  85  866   85  872  85  878 

85  884  85  890  85  896  85  902  85  908 

6 

723 

85  9H  85  920  85  926  85  932  85  938 

85  944  85  950  85  956  85  962  85  968 

6 

724 

85  974  85  980  85  986  85  992  85  998 

86004  86010  86016  86022  86028 

6 

725 

86034  86040  86046  86052  86058 

86064  86070  86076  86082  86088 

6 

726 

86  094  86  100  86  106  86  112  86  118 

86  124  86  130  86  136  86  141  86  147 

6 

727 

86  153  86  159  86  165  86  171  86  177 

86  183  86  189  86  195  86  201  86  207 

6 

728 

86213  86219  86225  86231  86237 

86  243  86  249  86  255  86  261  86  267 

6 

729 

86273  86279  86285  86291  86297 

86303   86308  86314  86320  86326 

6 

730 

86332  86338   86344  86350  86356 

86362   86368  86374  86380  86386 

6 

731 

86392  86398  86404  86410  86415 

86421  86427  86433   86439  86445 

6 

732 

86451  86457  86463  86469  86475 

86481  86487  86493  86499  86504 

6 

733 

86510  86516  86522  86528  86534 

86540  86546  86552  86558  86564 

6 

734 

86570  86576  86581  86587  86593 

86599  86605  86  611  86617  86623 

6 

735 

86629  86635  86641  86646   86652 

86658  86664   86670  86676  86682 

6 

736 

86  688  86  694  86  700  86  705  86  711 

86  717  86  723  86  729  86  735  86  741 

6 

737 

86  747  86  TS3   86  759  86  764  86  770 

86  776  86  782  86  788  86  794  86  800 

6 

738 

86806  86812  86817  86823   86829 

86835  86841  86847  86853  86859 

6 

739 

86864  86870  86876  86882  86  888 

86894  86900  86906  86  911  86917 

6 

740 

86923  86929  86935  86941  86947 

86953  86958  86964  86970  86976 

6 

741 

86  982  86  988  86  994  86  999  87  005 

87  on  87017  87023  87029  87035 

6 

742 

87  040  87  046  87  052  87  058  87  064 

87  070  87  075  87  081  87  087  87  093 

6 

743 

87  099  87  105  87  III  87  116  87  122 

87  128  87  134  87  140  87  146  87  151 

6 

744 

87  157  87  163  87  169  87  17s  87  181 

87  186  87  192  87  198  87  204  87  210 

6 

745 

87  216  87  221  87  227  87  233  87  239 

87  245  87  251  87  256  87  262  87  268 

6 

746 

87  274  87  280  87  286  87  291  87  297 

87303  87309  87315  87320  87326 

6 

747 

87332  87338   87344  87349  87355 

87361  87367  87373  87379  87384 

6 

748 

87  390  87  396  87  402  87  408  87  413 

87419  87425  87431  87437  87442 

6 

749 

87  448  87  454  87  460  8y  466  87  471 

87  477  87  483  87  489  87  495  87  500 

6 

750 

87506  87512  87518  87523  87529 

87535  87541  87547  87552  87558 

6 

N 

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5    6    7    8    9 

N 

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5    6     7    8    9 

D 

750 

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87535  87541  87547  87552  87558 

6 

751 

87564  87570  87576  87581  87587 

87  593  87  599  87  604  87  610  87  616 

6 

752 

87  622  87  628  87  633   87  639  87  645 

87651  87656  87662  87668  87674 

6 

753 

87679  87685  87691  87697  87703 

87  708  87  714  87  720  87  726  87  731 

6 

754 

87  737  87  743  87  749  87  754  87  760 

87766  87772  87777  87783  87789 

6 

755 

87795  87800  87806  87812  87818 

87823  87829  87835  87841  87846 

6 

756 

87  852  87  858  87  864  87  869  87  875 

87881  87887  87892  87898  87904 

6 

757 

87910  87915  87921  87927  87933 

87  938  87  944  87  950  87  955  87  961 

6 

758 

87  967  87  973  87  978  87  984  87  990 

87996  88001  88007  88013  88018 

6 

759 

88  024  88  030  88  036  88  041  88  047 

88  053  88  058  88  064  88  070  88  076 

6 

760 

88081  88087  88093  88098  88104 

88  no  88  116  88  121  88  127  88  133 

6 

761 

88  138  88  144  88  150  88  156  88  161 

88  167  88  173  88  178  88  184  88  190 

6 

762 

88195  88201  88207  88213  88218 

88  224  88  230  88  235  88  241  88  247 

6 

763 

88  252  88  258  88  264  88  270  88  275 

88281  88287  88292  88298  88304 

6 

764 

88309  88315  88321  88326  88332 

88  338   88  343  88  349  88  355  88  360 

6 

765 

88366  88372  88377  88383  88389 

88395  88400  88406  88412  88417 

6 

766 

88  423  88  429  88  434  88  440  88  446 

88451  88457  88463  88468  88474 

6 

767 

88480  88485  88491  88497  88502 

88508  88513  88519  88525  88530 

6 

768 

88536  88542  88547  88553  88559 

88564  88570  88576  88581  88587 

6 

769 

88593  88598  88604  88610  88615 

88621  88627  88632  88638  88643 

6 

770 

88  649  88  655  88  660  88  666  88  672 

88  677  88  683  88  689  88  694  88  700 

•  6 

771 

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772 

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6 

773 

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6 

774 

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6 

775 

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6 

776 

88  986  88  992  88  997  89  003  89  009 

89014  89020  89025  89031  89037 

6 

777 

89  042  89  048  89  053  89  059  89  064 

89070  89076  89081  89087  89092 

6 

778 

89  098  89  104  89  109  89  ii5  89  120 

89  126  89  131  89  137  89  143  89  148 

6 

779 

89  154  89  159  89  165  89  170  89  176 

89  182  89  187  89  193  8g  198  89  204 

6 

780 

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89  237  89  243  89  248  89  254  89  260 

6 

781 

89265  89271  89276  89282  89287 

89293  89298  89304  89310  89315 

6 

782 

89321  89326  89332  89337  89343 

89348  89354  89360  89365  89371 

6 

783 

89376  89382  89387  89393  89398  1  89404  89409  89415  89421  89426 

5+ 

784 

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89459  89465  89470  89476  89481 

5+ 

785 

89  487  89  492  89  498  89  504  89  509 

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90  064  90  069  90  075  90  080  90  086 

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91  004 

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91  036  91  041  91  046  91  052 

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91  777  91  782  91  787  91  793 

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92  505  92  511  92  516  92  521 

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92  681 

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92  967  92973 
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92  978  92  983  92  988 

93  029  93  034  93  039 
93  080  93  085  93  090 
93  131  93  136  93  141 
93  181  93  186  93  192 

5 
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93  399  93  304  93  409  93  4i4  93  420 

93222  93  227 
93  273  93  278 
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93  374  93  379 
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93  232  93  237  93  242 
93  283  93  288  93  293 
93  334  93  339  93  344 
93  384  93  389  93  394 
93  435  93  440  93  445 

5 
5 
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860 

861 
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863 
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93  450  93  455  93  460  93  465  93  470 
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93551  93556  93561  93566  93571 
93601  93606  93  611  93616  93621 
93651  93656  93661  93666  93671 

93  475  93480 
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93  485  93  490  93  495 
93  536  93  541  93  546 
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93  63^  93  641  93  646 
93  687  93  692  93  697 

5 
5 
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93  702  93  707  93  712  93  717  93  722 
93  752  93  757  93  762  93  767  93  772 
93802  93807  93812  93817  93822 
93852  93857  93862  93867  93872 
93902  93907  93912  93917  93922 

93  727  93  732 
93  777  93  782 
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93  737  93  742  93  747 
93  787  93  792  93  797 
93  837  93  842  93  847 
93  887  93  892  93  897 
93  937  93942  93  947 

5 
5 
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870 
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94052  94057  94062  94067  94072 
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94  151  94156  94  161  94166  94  171 

93  977  93  982 
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94  126  94  131 
94  176  94  181 

93  987  93  992  93  997 
94037  94042  94047 
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94  136  94  141  94  146 
94  186  94  191  94  196 

5 
5 
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94  349  94  354  94  359  94364  94369 
94  399  94404  94409  94414  94419 

94226  94231  94236  94240  94245 
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94  424  94  429  94  433  94  438  94  443 

5 
5 
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880 
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94  473  94478 
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94483  94488  94  493 
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94581  94586  94591 
94630  94635  94640 
94  680  94  685  94  689 

5 
5 
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885 
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94  743  94  748  94  753  94  758  94  763 
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94719  94724  94729  94  734  94738 
94768  94  773  94778  94783  94787 
94817  94822  94827  94832  94836 
94866  94871  94876  94880  94885 
94915  94919  94924  94929  94  934 

5 
5 
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94  988  94  993  94  998  95  002  95  007 

95  036  95  041  95  046  95  051  95  056 
95  085  95  090  95  095  95  100  95  105 
95  134  95  139  95  143  95  148  95  i53 

94963  94968  94  973  94978  94983 
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95061  95066  95071  95075  95080 
95  109  95  114  95  119  95  124  95  129 
95  158  95  163  95  168  95  173  95  177 

5 
5 

5 
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95  182  95  187  95  192  95  197  95  202 
95  231  95  236  95  240  95  245  95  250 
95  279  95  284  95  289  95  294  95  299 
95  328  95  332  95  337  95  342  95  347 
95  376  95  381  95  386  95  390  95  395 

95  207  95  211  95  216  95  221  95  226 

95  255  95  260  95  265  95  270  95  274 
95303  95308  95313  95318  95323 
95352  95357  95361  95366  95371 
95  400  95  405  95  410  95  415  95  419 

5 
5 
5 
5 
5 

900 

95  424  95  429  95  434  95  439  95  444 

95  448  95  453 

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95  497  95  501  95  506  95  511  95  5^6 

5 

902 

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95  545  95  550  95  554  95  559  95  564 

5 

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95  569  95  574  95  578  95  583  95  588 

95  593  95  598  95  602  95  607  95  612 

5 

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95  641  95  646  95  650  95  655  95  660 

5 

90s 

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95  689  95  694  95  698  95  703  95  708 

5 

906 

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95  737  95  742  95  746  95  751  95  756 

5 

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95  761  95  766  95  770  95  775  95  780 

95  785  95  789  95  794  95  799  95  804 

5 

908 

95  809  95  813  95  818  95  823  95  828 

95  832  95  837  95  842  95  847  95  852 

5 

909 

95  856  95  861  95  866  95  871  95  875 

95  880  95  885  95  890  95  895  95  899 

5 

910 

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95  928  95  933  95  938  95  942  95  947 

5 

911 

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95  976  95  980  95  985  95  99o  95  995 

5 

9T2 

95  999  96  004  96  009  96  014  96  019 

96  023  96  028  96  033  96  038  96  042 

5 

913 

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96071  96076  96080  96085  96090 

5 

914 

96  095  96  099  96  104  96  109  96  114 

96  118  96  123  96  128  96  133  96  137 

5 

915 

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96  166  96  171  96  175  96  180  96  185 

5 

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5 

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96  261  96  265  96  270  96  275  96  280 

5 

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5 

920 

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5 

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5 

922 

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5 

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96544  96548  96553  96558  96562 

5 

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5 

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5 

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96  731  96  736  96  741  96  745  96  750 

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96  778  96  783  96  788  96  792  96  797 

5 

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96  825  96  830  96  834  96  839  96  844 

5 

930 

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5 

931 

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5 

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96965  96970  96974  96979  96984 

5 

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97  104  97  109  97  114  97  118  97  123 

5 

936 

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97  151  97  155  97  160  97  165  97  169 

5 

937 

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97  197  97  202  97  206  97  211  97  216 

5 

938 

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97  243  97  248  97  253  97  257  97  262 

5 

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97  290  97  294  97  299  97  304  97  308 

5 

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5 

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97  474  97  479  97  483  97  488  97  493 

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5 

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97  612  97  617  97  621  97  626  97  630 

5 

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5 

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97  749  97  754  97  759  97  763  97  768 

5 

950 

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97  795  97  800  97  804  97  809  97  813 

5 

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5    6    7    8    9 

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D 

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97  795  97  800  97  804  97  809  97  813 

5 

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97818  97823  97827  97832  97836 

97841  97845  97850  97855  97859 

5 

952 

97  864  97  868  97  873  97  877  97  882 

97  886  97  891  97  896  97  900  97  905 

5 

953 

97  909  97  914  97  918  97  923  97  928 

97932  97  937  97941  97946  97950 

5 

954 

97  955  97  959  97  9^4  97  9'^^   97  973 

97978  97982  97987  97991  97996 

5 

955 

98000  98005  98009  98014  98019 

98  023  98  028  98  032  98  037  98  041 

4+ 

956 

98  046  98  050  98  055  98  059  98  064 

98068  98073  98078  98082  98087 

4+ 

957 

98  091  98  096  98  100  98  io5  98  109 

98  114  98  118  98  123  98  127  98  132 

4+ 

95S 

98  137  98  141  98  146  98  150  98  155 

98  159  98  164  98  168  98  173  98  177 

4+ 

959 

98  182  98  186  98  191  98  195  98  200 

98  204  98  209  98  214  98  218  98  223 

44- 

960 

98  227  98  232  98  236  98  241  98  245 

98  250  98  254  98  259  98  263  98  268 

4+ 

961 

98  272  98  277  98  281  98  286  98  290 

98295  98299  98304  98308  98313 

4+ 

962 

98318  98322  98327  98331  98336 

98340  98345  98349  98354  98358 

4+ 

963 

98363  98367  98372  98376  98381 

98  385  98  Z9°   98  394  9^  399  98  403 

4+ 

964 

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98  430  98  435  98  439  98  444  98  448 

4+ 

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98  475  98  480  98  484  98  489  98  493 

4+ 

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98520  98525  98529  98534  98538 

4+ 

967 

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98565  98570  98574  98579  98583 

4+ 

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98610  98614  98619  98623  98628 

4+ 

969 

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98  655  98  659  98  664  98  668  98  673 

4+ 

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98  700  98  704  98  709  98  713  98  717 

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98  744  98  749  98  753  98  758  98  762 

4+ 

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98  789  98  793  98  798  98  802  98  807 

4+ 

973 

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98834  98838,98843  98847  98851 

4+ 

974 

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98878  98883  98887  98892  98896 

4+ 

975 

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98923  98927  98932  98936  98941 

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976 

98  945  98  949  98  954  98  958  98  963 

98967  98972  98976  98981  98985 

4 

977 

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99012  99016  99021  99025  99029 

4 

978 

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4 

979 

99  078  99  083  99  087  99  092  99  096 

99  100  99  io5  99  109  99  114  99  118 

4 

980 

99  123  99  127  99  131  99  136  99  140 

99  145  99  149  99  154  99  158  99  162 

4 

981 

99  167  99  171  99  176  99  180  99  185 

99  189  99  193  99  198  99  202  99  207 

4 

982 

99  211  99  216  99  220  99  224  99  229 

99  233  99  238  99  242  99  247  99  251 

4 

983 

99  255  99  260  99  264  99  269  99  273 

99277  99282  99286  99291  99295 

4 

984 

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99322  99326  99330  99335  99ZZ9 

4 

98s 

99  344  99348  99352  99357  99361 

99366  99370  99  374  99  379  99383 

4 

986 

99388  99392  99396  99401  99405 

99410  99414  99419  99423  99427 

4 

987 

99  432  99  436  99  441  99  445  99  449 

99  454  99458  99463  99467  99471 

4 

988 

99  476  99  480  99  484  99  489  99  493 

99498  99502  99506  99  511  99515 

4 

989 

99520  99524  99528  99533  99.537 

99542  99546  99550  99555  99  559 

4 

990 

99564  99568  99572  99577  99581 

,  99  585  99  590  99  594  99  599  99  603 

4 

991 

99607  99612  99616  99621  99625 

99  629  99  634  99  638  99  642  99  647 

4 

992 

99651  99656  99660  99664  99669 

99673  99677  99682  99686  99691 

4 

993 

99  695  99  699  99  704  99  708  99  712 

99717  99721  99726  99730  99  734 

4 

994 

99  739  99  743  99  747  99  752  99  756 

99  760  99  765  99  769  99  774  99  778 

■  4 

995 

99782  99787  99791  99  795  99800 

99804  99808  99813  99817  99822 

4 

996 

99826  99830  99835  99839  99843 

99848  99852  99856  99861  99865 

4 

997 

99870  99874  99878  99883  99887 

99891  99896  99900  99904  99909 

4 

998 

99913  99917  99922  99926  99930 

99  935  99  939  99  944  99  948  99  952 

4 

999 

99  957  99961  99965  99970  99974 

99978  99983  99987  99991  99996 

4 

1000 

00000  00004  00009  00013  00017 

00  022  00  026  00  030  00  035  00  039 

4 

N 

0    12    3    4 

5    6    7    8    9 

Answers. 


Page  4. 

II. 

276 

3. 

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Page  12. 

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Page  10. 

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3- 

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to.     8.25 

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4. 

5-21 

^ 

-vi 


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(N8837sl0)476 — A-32  Berkeley 


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